LMFDB - Number field 45.45.2068824372668549386552025657117917020136846925000101250046771586030929014642966434767146211015642620623111724853515625.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251)

gp: K = bnfinit(y^45 - 15*y^44 - 120*y^43 + 2785*y^42 + 2040*y^41 - 222342*y^40 + 420425*y^39 + 10024590*y^38 - 35069730*y^37 - 280556895*y^36 + 1389459441*y^35 + 4980489270*y^34 - 34327919505*y^33 - 52485204525*y^32 + 576080544270*y^31 + 196092367854*y^30 - 6820772763030*y^29 + 3051952728270*y^28 + 57922300812880*y^27 - 59937846525180*y^26 - 353495076875202*y^25 + 542694665981220*y^24 + 1531071286376460*y^23 - 3131712122030760*y^22 - 4545064586341055*y^21 + 12339167509476885*y^20 + 8461213880546430*y^19 - 33709107574234355*y^18 - 6968642817313350*y^17 + 63373889569191960*y^16 - 7076452922900098*y^15 - 80218608988132905*y^14 + 26470314281644410*y^13 + 66243722166245240*y^12 - 32060240741785995*y^11 - 34356066621763764*y^10 + 20781515138515650*y^9 + 10477766738441505*y^8 - 7675031336820660*y^7 - 1594722107742745*y^6 + 1559444254090284*y^5 + 49643486026125*y^4 - 150881459933485*y^3 + 10931860799430*y^2 + 4297070637735*y - 401780151251, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251)

$ x^{45} - 15 x^{44} - 120 x^{43} + 2785 x^{42} + 2040 x^{41} - 222342 x^{40} + 420425 x^{39} + \cdots - 401780151251 $ x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$45$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[45, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$206\!\cdots\!625$ 2068824372668549386552025657117917020136846925000101250046771586030929014642966434767146211015642620623111724853515625 $\medspace = 3^{60}\cdot 5^{72}\cdot 19^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$404.59$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{4/3}5^{8/5}19^{2/3}\approx 404.59087212305565$
Ramified primes:	$3$, $5$, $19$ 3, 5, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$45$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$4275=3^{2}\cdot 5^{2}\cdot 19$
Dirichlet character group:	$\lbrace$$\chi_{4275}(4096,·)$, $\chi_{4275}(1,·)$, $\chi_{4275}(2566,·)$, $\chi_{4275}(391,·)$, $\chi_{4275}(1546,·)$, $\chi_{4275}(2956,·)$, $\chi_{4275}(4111,·)$, $\chi_{4275}(1426,·)$, $\chi_{4275}(406,·)$, $\chi_{4275}(3991,·)$, $\chi_{4275}(1816,·)$, $\chi_{4275}(2971,·)$, $\chi_{4275}(286,·)$, $\chi_{4275}(2851,·)$, $\chi_{4275}(676,·)$, $\chi_{4275}(1831,·)$, $\chi_{4275}(3241,·)$, $\chi_{4275}(1711,·)$, $\chi_{4275}(691,·)$, $\chi_{4275}(2101,·)$, $\chi_{4275}(3256,·)$, $\chi_{4275}(571,·)$, $\chi_{4275}(3136,·)$, $\chi_{4275}(961,·)$, $\chi_{4275}(2116,·)$, $\chi_{4275}(3526,·)$, $\chi_{4275}(1996,·)$, $\chi_{4275}(976,·)$, $\chi_{4275}(2386,·)$, $\chi_{4275}(3541,·)$, $\chi_{4275}(856,·)$, $\chi_{4275}(3421,·)$, $\chi_{4275}(1246,·)$, $\chi_{4275}(2401,·)$, $\chi_{4275}(3811,·)$, $\chi_{4275}(2281,·)$, $\chi_{4275}(106,·)$, $\chi_{4275}(1261,·)$, $\chi_{4275}(2671,·)$, $\chi_{4275}(3826,·)$, $\chi_{4275}(1141,·)$, $\chi_{4275}(121,·)$, $\chi_{4275}(3706,·)$, $\chi_{4275}(1531,·)$, $\chi_{4275}(2686,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{14}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{2}{7}a^{15}+\frac{3}{14}a^{9}+\frac{5}{14}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{14}a^{3}-\frac{1}{2}a^{2}-\frac{5}{14}a+\frac{1}{7}$, $\frac{1}{14}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{3}{14}a^{16}+\frac{3}{14}a^{10}-\frac{1}{7}a^{8}-\frac{1}{2}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{2}-\frac{5}{14}a-\frac{1}{2}$, $\frac{1}{14}a^{23}+\frac{2}{7}a^{17}-\frac{1}{2}a^{16}+\frac{3}{14}a^{11}-\frac{1}{7}a^{9}-\frac{1}{2}a^{8}+\frac{3}{7}a^{5}-\frac{1}{2}a^{4}-\frac{5}{14}a^{3}+\frac{1}{7}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{24}+\frac{2}{7}a^{18}-\frac{1}{2}a^{17}+\frac{3}{14}a^{12}-\frac{1}{7}a^{10}-\frac{1}{2}a^{9}+\frac{3}{7}a^{6}-\frac{1}{2}a^{5}-\frac{5}{14}a^{4}+\frac{1}{7}a^{3}-\frac{1}{2}a$, $\frac{1}{14}a^{25}+\frac{2}{7}a^{19}-\frac{1}{2}a^{18}+\frac{3}{14}a^{13}-\frac{1}{7}a^{11}-\frac{1}{2}a^{10}+\frac{3}{7}a^{7}-\frac{1}{2}a^{6}-\frac{5}{14}a^{5}+\frac{1}{7}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{26}-\frac{3}{14}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{3}{14}a^{14}-\frac{1}{7}a^{12}-\frac{1}{2}a^{11}-\frac{1}{14}a^{8}-\frac{1}{2}a^{7}+\frac{1}{7}a^{6}+\frac{1}{7}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{14}a^{27}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{14}a^{15}-\frac{1}{7}a^{13}-\frac{1}{2}a^{12}-\frac{3}{7}a^{9}+\frac{3}{14}a^{7}-\frac{5}{14}a^{6}-\frac{3}{14}a^{3}-\frac{1}{2}a^{2}-\frac{1}{14}a-\frac{1}{14}$, $\frac{1}{14}a^{28}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{14}a^{16}-\frac{1}{7}a^{14}-\frac{1}{2}a^{13}-\frac{3}{7}a^{10}+\frac{3}{14}a^{8}-\frac{5}{14}a^{7}-\frac{3}{14}a^{4}-\frac{1}{2}a^{3}-\frac{1}{14}a^{2}-\frac{1}{14}a$, $\frac{1}{14}a^{29}-\frac{1}{2}a^{19}-\frac{3}{7}a^{17}-\frac{1}{2}a^{16}-\frac{1}{7}a^{15}-\frac{1}{2}a^{14}-\frac{3}{7}a^{11}+\frac{3}{14}a^{9}+\frac{1}{7}a^{8}-\frac{1}{2}a^{6}-\frac{3}{14}a^{5}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{350}a^{30}-\frac{1}{35}a^{29}+\frac{1}{70}a^{28}-\frac{1}{35}a^{27}+\frac{1}{35}a^{26}+\frac{6}{175}a^{25}+\frac{1}{35}a^{23}-\frac{1}{70}a^{22}+\frac{4}{35}a^{20}+\frac{16}{35}a^{19}-\frac{2}{35}a^{18}+\frac{3}{35}a^{17}+\frac{33}{70}a^{16}-\frac{67}{350}a^{15}+\frac{16}{35}a^{14}-\frac{1}{10}a^{13}-\frac{11}{35}a^{12}-\frac{13}{35}a^{11}+\frac{11}{25}a^{10}-\frac{17}{35}a^{9}+\frac{17}{35}a^{8}+\frac{8}{35}a^{7}-\frac{1}{35}a^{6}-\frac{111}{350}a^{5}+\frac{9}{35}a^{4}-\frac{13}{35}a^{3}+\frac{17}{70}a^{2}+\frac{2}{7}a-\frac{16}{175}$, $\frac{1}{350}a^{31}+\frac{1}{70}a^{29}-\frac{1}{35}a^{28}+\frac{1}{35}a^{27}+\frac{6}{175}a^{26}-\frac{1}{70}a^{25}+\frac{1}{35}a^{24}-\frac{1}{70}a^{23}-\frac{1}{35}a^{21}-\frac{3}{70}a^{20}+\frac{3}{35}a^{19}-\frac{17}{35}a^{18}-\frac{1}{35}a^{17}+\frac{79}{175}a^{16}-\frac{11}{35}a^{15}-\frac{1}{10}a^{14}+\frac{3}{70}a^{13}+\frac{2}{35}a^{12}-\frac{23}{175}a^{11}-\frac{3}{10}a^{10}-\frac{3}{35}a^{9}-\frac{19}{70}a^{8}-\frac{1}{35}a^{7}+\frac{139}{350}a^{6}-\frac{19}{70}a^{5}+\frac{19}{70}a^{4}+\frac{16}{35}a^{3}+\frac{2}{7}a^{2}+\frac{43}{350}a+\frac{1}{70}$, $\frac{1}{350}a^{32}-\frac{1}{35}a^{29}+\frac{1}{35}a^{28}+\frac{6}{175}a^{27}-\frac{1}{70}a^{26}-\frac{1}{70}a^{24}-\frac{1}{35}a^{22}+\frac{1}{35}a^{21}+\frac{3}{35}a^{20}+\frac{3}{10}a^{19}+\frac{9}{35}a^{18}-\frac{17}{350}a^{17}-\frac{27}{70}a^{16}+\frac{2}{7}a^{15}+\frac{3}{70}a^{14}-\frac{8}{35}a^{13}+\frac{27}{175}a^{12}-\frac{31}{70}a^{11}+\frac{1}{14}a^{10}-\frac{17}{35}a^{9}+\frac{33}{70}a^{8}-\frac{111}{350}a^{7}-\frac{9}{70}a^{6}-\frac{2}{7}a^{5}+\frac{11}{35}a^{4}+\frac{3}{7}a^{3}-\frac{66}{175}a^{2}-\frac{12}{35}a+\frac{17}{70}$, $\frac{1}{2450}a^{33}+\frac{1}{1225}a^{32}+\frac{3}{2450}a^{31}-\frac{1}{35}a^{29}+\frac{1}{1225}a^{28}+\frac{1}{50}a^{27}+\frac{13}{1225}a^{26}+\frac{1}{98}a^{25}-\frac{11}{490}a^{24}+\frac{1}{49}a^{23}-\frac{1}{70}a^{22}+\frac{9}{490}a^{21}+\frac{12}{245}a^{20}+\frac{93}{490}a^{19}+\frac{79}{350}a^{18}+\frac{538}{1225}a^{17}+\frac{79}{2450}a^{16}+\frac{153}{490}a^{15}-\frac{68}{245}a^{14}-\frac{418}{1225}a^{13}+\frac{319}{1225}a^{12}+\frac{151}{1225}a^{11}+\frac{11}{490}a^{10}+\frac{26}{245}a^{9}+\frac{709}{2450}a^{8}-\frac{73}{175}a^{7}+\frac{27}{2450}a^{6}-\frac{31}{245}a^{5}-\frac{72}{245}a^{4}-\frac{76}{1225}a^{3}-\frac{81}{175}a^{2}-\frac{13}{1225}a+\frac{69}{245}$, $\frac{1}{2450}a^{34}-\frac{1}{2450}a^{32}+\frac{1}{2450}a^{31}+\frac{1}{1225}a^{29}-\frac{1}{98}a^{28}-\frac{1}{1225}a^{27}+\frac{57}{2450}a^{26}+\frac{11}{490}a^{24}+\frac{1}{490}a^{23}-\frac{6}{245}a^{22}-\frac{4}{245}a^{21}+\frac{12}{245}a^{20}+\frac{883}{2450}a^{19}-\frac{87}{245}a^{18}-\frac{393}{2450}a^{17}+\frac{69}{1225}a^{16}-\frac{67}{245}a^{15}-\frac{473}{1225}a^{14}-\frac{31}{70}a^{13}-\frac{309}{2450}a^{12}-\frac{523}{1225}a^{11}+\frac{92}{245}a^{10}-\frac{3}{350}a^{9}+\frac{57}{245}a^{8}-\frac{99}{2450}a^{7}+\frac{6}{175}a^{6}+\frac{109}{245}a^{5}-\frac{171}{350}a^{4}-\frac{117}{490}a^{3}-\frac{279}{1225}a^{2}-\frac{38}{175}a-\frac{61}{245}$, $\frac{1}{2450}a^{35}+\frac{3}{2450}a^{32}+\frac{3}{2450}a^{31}+\frac{1}{1225}a^{30}+\frac{8}{245}a^{29}-\frac{69}{2450}a^{27}+\frac{13}{1225}a^{26}+\frac{8}{245}a^{25}-\frac{1}{49}a^{24}-\frac{1}{245}a^{23}-\frac{3}{98}a^{22}-\frac{1}{245}a^{21}-\frac{111}{1225}a^{20}-\frac{81}{490}a^{19}-\frac{213}{490}a^{18}-\frac{1061}{2450}a^{17}+\frac{317}{1225}a^{16}+\frac{522}{1225}a^{15}-\frac{54}{245}a^{14}-\frac{159}{490}a^{13}+\frac{817}{2450}a^{12}+\frac{86}{1225}a^{11}+\frac{17}{1225}a^{10}-\frac{57}{245}a^{9}-\frac{53}{490}a^{8}+\frac{8}{175}a^{7}-\frac{229}{1225}a^{6}-\frac{807}{2450}a^{5}+\frac{229}{490}a^{4}-\frac{37}{490}a^{3}+\frac{87}{350}a^{2}+\frac{207}{1225}a+\frac{103}{490}$, $\frac{1}{12250}a^{36}+\frac{1}{6125}a^{35}-\frac{1}{12250}a^{34}-\frac{1}{12250}a^{33}+\frac{1}{6125}a^{32}+\frac{8}{6125}a^{31}-\frac{1}{1750}a^{30}-\frac{61}{6125}a^{29}+\frac{333}{12250}a^{28}-\frac{61}{12250}a^{27}+\frac{13}{12250}a^{26}+\frac{194}{6125}a^{25}+\frac{53}{2450}a^{24}+\frac{17}{2450}a^{23}-\frac{3}{1225}a^{22}-\frac{417}{12250}a^{21}-\frac{186}{875}a^{20}-\frac{1223}{12250}a^{19}-\frac{1304}{6125}a^{18}+\frac{1518}{6125}a^{17}+\frac{68}{6125}a^{16}+\frac{347}{1225}a^{15}-\frac{4929}{12250}a^{14}-\frac{5969}{12250}a^{13}+\frac{2459}{6125}a^{12}-\frac{292}{1225}a^{11}+\frac{2439}{12250}a^{10}+\frac{433}{6125}a^{9}+\frac{3561}{12250}a^{8}-\frac{1087}{12250}a^{7}-\frac{2111}{12250}a^{6}-\frac{5163}{12250}a^{5}-\frac{2194}{6125}a^{4}+\frac{2536}{6125}a^{3}+\frac{1683}{6125}a^{2}+\frac{1966}{6125}a-\frac{302}{875}$, $\frac{1}{12250}a^{37}+\frac{1}{12250}a^{34}-\frac{1}{12250}a^{33}+\frac{17}{12250}a^{32}-\frac{2}{6125}a^{31}+\frac{1}{1750}a^{30}-\frac{423}{12250}a^{29}+\frac{313}{12250}a^{28}-\frac{4}{175}a^{27}+\frac{41}{6125}a^{26}-\frac{13}{6125}a^{25}-\frac{1}{175}a^{24}+\frac{3}{98}a^{23}-\frac{207}{12250}a^{22}+\frac{23}{1225}a^{21}-\frac{3}{35}a^{20}-\frac{1206}{6125}a^{19}+\frac{2181}{6125}a^{18}+\frac{3679}{12250}a^{17}+\frac{1289}{6125}a^{16}+\frac{2008}{6125}a^{15}-\frac{5211}{12250}a^{14}+\frac{3411}{12250}a^{13}-\frac{418}{6125}a^{12}-\frac{631}{12250}a^{11}-\frac{186}{6125}a^{10}+\frac{3279}{12250}a^{9}+\frac{2321}{12250}a^{8}-\frac{328}{875}a^{7}+\frac{2412}{6125}a^{6}+\frac{4223}{12250}a^{5}-\frac{3026}{6125}a^{4}+\frac{2566}{6125}a^{3}-\frac{9}{25}a^{2}+\frac{619}{6125}a-\frac{71}{250}$, $\frac{1}{12250}a^{38}+\frac{1}{12250}a^{35}-\frac{1}{12250}a^{34}+\frac{1}{6125}a^{33}+\frac{1}{12250}a^{32}-\frac{3}{12250}a^{31}-\frac{3}{12250}a^{30}-\frac{387}{12250}a^{29}+\frac{4}{1225}a^{28}+\frac{146}{6125}a^{27}-\frac{173}{6125}a^{26}+\frac{9}{2450}a^{25}-\frac{3}{98}a^{24}-\frac{216}{6125}a^{23}+\frac{11}{2450}a^{22}+\frac{1}{490}a^{21}+\frac{869}{6125}a^{20}+\frac{1406}{6125}a^{19}-\frac{4441}{12250}a^{18}+\frac{5093}{12250}a^{17}-\frac{1439}{12250}a^{16}-\frac{2301}{12250}a^{15}+\frac{3111}{12250}a^{14}-\frac{27}{125}a^{13}+\frac{2407}{6125}a^{12}-\frac{2206}{6125}a^{11}+\frac{3959}{12250}a^{10}-\frac{177}{6125}a^{9}-\frac{876}{6125}a^{8}-\frac{1231}{12250}a^{7}-\frac{1467}{12250}a^{6}+\frac{2903}{12250}a^{5}-\frac{212}{875}a^{4}+\frac{1149}{2450}a^{3}-\frac{547}{12250}a^{2}+\frac{3841}{12250}a+\frac{597}{2450}$, $\frac{1}{1849750}a^{39}+\frac{3}{924875}a^{38}+\frac{71}{1849750}a^{37}-\frac{71}{1849750}a^{36}+\frac{181}{1849750}a^{35}-\frac{163}{924875}a^{34}+\frac{157}{924875}a^{33}-\frac{549}{1849750}a^{32}-\frac{1847}{1849750}a^{31}-\frac{1599}{1849750}a^{30}+\frac{59589}{1849750}a^{29}-\frac{1614}{132125}a^{28}-\frac{30087}{1849750}a^{27}-\frac{817}{36995}a^{26}+\frac{7999}{264250}a^{25}+\frac{18584}{924875}a^{24}-\frac{27857}{1849750}a^{23}+\frac{40543}{1849750}a^{22}+\frac{421}{924875}a^{21}+\frac{201069}{924875}a^{20}+\frac{50644}{184975}a^{19}-\frac{5295}{14798}a^{18}-\frac{689809}{1849750}a^{17}+\frac{884441}{1849750}a^{16}+\frac{41093}{264250}a^{15}-\frac{685333}{1849750}a^{14}+\frac{15133}{132125}a^{13}-\frac{112673}{369950}a^{12}+\frac{18759}{37750}a^{11}+\frac{4056}{26425}a^{10}+\frac{12714}{132125}a^{9}+\frac{246203}{924875}a^{8}+\frac{412497}{924875}a^{7}+\frac{159056}{924875}a^{6}-\frac{433218}{924875}a^{5}-\frac{775939}{1849750}a^{4}-\frac{333482}{924875}a^{3}+\frac{35001}{924875}a^{2}+\frac{177511}{369950}a+\frac{50331}{924875}$, $\frac{1}{606040991500}a^{40}+\frac{9091}{60604099150}a^{39}+\frac{920781}{151510247875}a^{38}+\frac{1302657}{151510247875}a^{37}+\frac{13478903}{606040991500}a^{36}+\frac{24428647}{303020495750}a^{35}-\frac{8413093}{121208198300}a^{34}-\frac{58495473}{303020495750}a^{33}-\frac{818509317}{606040991500}a^{32}+\frac{151275571}{303020495750}a^{31}-\frac{33119853}{303020495750}a^{30}+\frac{40844813}{1731545690}a^{29}-\frac{1013678762}{151510247875}a^{28}-\frac{528051212}{21644321125}a^{27}-\frac{4158481912}{151510247875}a^{26}+\frac{6521589719}{303020495750}a^{25}-\frac{372340641}{30302049575}a^{24}+\frac{6327643111}{303020495750}a^{23}+\frac{286405683}{21644321125}a^{22}+\frac{1970641077}{303020495750}a^{21}-\frac{6795016999}{60604099150}a^{20}-\frac{21588964}{123681835}a^{19}-\frac{22494657309}{151510247875}a^{18}-\frac{4833311913}{43288642250}a^{17}-\frac{55573513873}{121208198300}a^{16}+\frac{45787456098}{151510247875}a^{15}-\frac{12133922328}{30302049575}a^{14}-\frac{9613254138}{30302049575}a^{13}-\frac{10139366183}{24241639660}a^{12}-\frac{8931025276}{151510247875}a^{11}+\frac{113290147073}{303020495750}a^{10}-\frac{18607240299}{60604099150}a^{9}+\frac{198714302831}{606040991500}a^{8}+\frac{96043037791}{303020495750}a^{7}-\frac{5478157681}{86577284500}a^{6}+\frac{18940543149}{151510247875}a^{5}-\frac{8298452499}{17315456900}a^{4}-\frac{34327164191}{303020495750}a^{3}-\frac{133549729239}{606040991500}a^{2}-\frac{32422606813}{151510247875}a-\frac{47998872399}{121208198300}$, $\frac{1}{606040991500}a^{41}+\frac{187}{6060409915}a^{39}-\frac{72354}{21644321125}a^{38}-\frac{719567}{121208198300}a^{37}-\frac{133963}{12120819830}a^{36}+\frac{23381189}{606040991500}a^{35}+\frac{8393647}{151510247875}a^{34}+\frac{1325411}{86577284500}a^{33}+\frac{89318109}{303020495750}a^{32}+\frac{50468422}{151510247875}a^{31}-\frac{22359226}{21644321125}a^{30}+\frac{295915964}{151510247875}a^{29}+\frac{3057727341}{303020495750}a^{28}-\frac{3557321426}{151510247875}a^{27}+\frac{6881114131}{303020495750}a^{26}+\frac{382192369}{303020495750}a^{25}-\frac{11546187}{865772845}a^{24}+\frac{160689703}{21644321125}a^{23}-\frac{951096079}{30302049575}a^{22}-\frac{1978765913}{151510247875}a^{21}-\frac{10437793077}{43288642250}a^{20}+\frac{53296929191}{151510247875}a^{19}+\frac{3005873588}{6060409915}a^{18}+\frac{45701950319}{606040991500}a^{17}-\frac{135232171149}{303020495750}a^{16}-\frac{44280446611}{303020495750}a^{15}+\frac{75394032423}{151510247875}a^{14}+\frac{42220214143}{86577284500}a^{13}+\frac{35285183351}{303020495750}a^{12}-\frac{4628171819}{21644321125}a^{11}+\frac{110006041681}{303020495750}a^{10}-\frac{5951837299}{86577284500}a^{9}-\frac{51795066251}{303020495750}a^{8}-\frac{141730455163}{606040991500}a^{7}+\frac{8656466703}{43288642250}a^{6}-\frac{24876124257}{606040991500}a^{5}+\frac{3485296456}{151510247875}a^{4}-\frac{109316218647}{606040991500}a^{3}+\frac{12079059186}{151510247875}a^{2}+\frac{29704633201}{606040991500}a+\frac{63844060038}{151510247875}$, $\frac{1}{73\!\cdots\!00}a^{42}-\frac{50\!\cdots\!57}{73\!\cdots\!00}a^{41}+\frac{28\!\cdots\!83}{36\!\cdots\!50}a^{40}+\frac{19\!\cdots\!91}{26\!\cdots\!75}a^{39}-\frac{11\!\cdots\!59}{73\!\cdots\!00}a^{38}+\frac{28\!\cdots\!69}{73\!\cdots\!00}a^{37}-\frac{16\!\cdots\!93}{73\!\cdots\!00}a^{36}+\frac{54\!\cdots\!17}{73\!\cdots\!00}a^{35}-\frac{45\!\cdots\!11}{73\!\cdots\!00}a^{34}-\frac{13\!\cdots\!17}{73\!\cdots\!00}a^{33}-\frac{24\!\cdots\!34}{36\!\cdots\!25}a^{32}-\frac{37\!\cdots\!71}{10\!\cdots\!50}a^{31}-\frac{84\!\cdots\!73}{36\!\cdots\!50}a^{30}-\frac{72\!\cdots\!43}{26\!\cdots\!75}a^{29}-\frac{22\!\cdots\!97}{18\!\cdots\!25}a^{28}+\frac{29\!\cdots\!24}{36\!\cdots\!25}a^{27}+\frac{95\!\cdots\!69}{36\!\cdots\!25}a^{26}-\frac{73\!\cdots\!51}{18\!\cdots\!25}a^{25}+\frac{39\!\cdots\!11}{18\!\cdots\!25}a^{24}-\frac{24\!\cdots\!68}{18\!\cdots\!25}a^{23}-\frac{10\!\cdots\!57}{14\!\cdots\!30}a^{22}+\frac{10\!\cdots\!46}{52\!\cdots\!75}a^{21}-\frac{58\!\cdots\!39}{18\!\cdots\!25}a^{20}+\frac{45\!\cdots\!59}{18\!\cdots\!25}a^{19}+\frac{91\!\cdots\!27}{73\!\cdots\!00}a^{18}+\frac{34\!\cdots\!79}{14\!\cdots\!00}a^{17}-\frac{47\!\cdots\!48}{36\!\cdots\!25}a^{16}+\frac{73\!\cdots\!54}{18\!\cdots\!25}a^{15}+\frac{38\!\cdots\!57}{10\!\cdots\!00}a^{14}-\frac{39\!\cdots\!97}{73\!\cdots\!00}a^{13}+\frac{20\!\cdots\!67}{73\!\cdots\!65}a^{12}-\frac{16\!\cdots\!67}{10\!\cdots\!50}a^{11}+\frac{17\!\cdots\!59}{73\!\cdots\!00}a^{10}-\frac{90\!\cdots\!19}{73\!\cdots\!00}a^{9}-\frac{21\!\cdots\!73}{73\!\cdots\!00}a^{8}+\frac{14\!\cdots\!69}{42\!\cdots\!80}a^{7}+\frac{37\!\cdots\!27}{14\!\cdots\!00}a^{6}-\frac{36\!\cdots\!59}{73\!\cdots\!00}a^{5}-\frac{22\!\cdots\!99}{15\!\cdots\!00}a^{4}-\frac{32\!\cdots\!53}{15\!\cdots\!00}a^{3}+\frac{51\!\cdots\!89}{10\!\cdots\!00}a^{2}-\frac{50\!\cdots\!51}{73\!\cdots\!00}a-\frac{13\!\cdots\!39}{36\!\cdots\!50}$, $\frac{1}{73\!\cdots\!00}a^{43}+\frac{60\!\cdots\!41}{73\!\cdots\!00}a^{41}+\frac{52\!\cdots\!81}{14\!\cdots\!00}a^{40}-\frac{18\!\cdots\!29}{73\!\cdots\!00}a^{39}+\frac{61\!\cdots\!39}{36\!\cdots\!50}a^{38}-\frac{46\!\cdots\!09}{73\!\cdots\!50}a^{37}+\frac{17\!\cdots\!63}{73\!\cdots\!00}a^{36}-\frac{50\!\cdots\!97}{36\!\cdots\!50}a^{35}+\frac{13\!\cdots\!13}{73\!\cdots\!00}a^{34}-\frac{10\!\cdots\!89}{14\!\cdots\!00}a^{33}-\frac{94\!\cdots\!39}{73\!\cdots\!00}a^{32}+\frac{31\!\cdots\!81}{36\!\cdots\!50}a^{31}-\frac{20\!\cdots\!68}{18\!\cdots\!25}a^{30}+\frac{12\!\cdots\!21}{36\!\cdots\!50}a^{29}+\frac{41\!\cdots\!97}{36\!\cdots\!50}a^{28}-\frac{62\!\cdots\!27}{18\!\cdots\!25}a^{27}-\frac{73\!\cdots\!11}{36\!\cdots\!50}a^{26}-\frac{10\!\cdots\!19}{36\!\cdots\!50}a^{25}+\frac{37\!\cdots\!99}{36\!\cdots\!50}a^{24}-\frac{45\!\cdots\!52}{18\!\cdots\!25}a^{23}+\frac{58\!\cdots\!88}{36\!\cdots\!25}a^{22}-\frac{10\!\cdots\!32}{18\!\cdots\!25}a^{21}-\frac{32\!\cdots\!38}{18\!\cdots\!25}a^{20}+\frac{69\!\cdots\!47}{73\!\cdots\!00}a^{19}+\frac{87\!\cdots\!63}{18\!\cdots\!25}a^{18}+\frac{30\!\cdots\!09}{10\!\cdots\!00}a^{17}-\frac{54\!\cdots\!07}{73\!\cdots\!00}a^{16}-\frac{34\!\cdots\!13}{73\!\cdots\!00}a^{15}+\frac{26\!\cdots\!27}{18\!\cdots\!25}a^{14}+\frac{33\!\cdots\!49}{73\!\cdots\!00}a^{13}-\frac{49\!\cdots\!91}{73\!\cdots\!00}a^{12}-\frac{20\!\cdots\!73}{73\!\cdots\!00}a^{11}+\frac{17\!\cdots\!09}{36\!\cdots\!50}a^{10}+\frac{39\!\cdots\!04}{26\!\cdots\!75}a^{9}+\frac{27\!\cdots\!69}{14\!\cdots\!00}a^{8}+\frac{67\!\cdots\!26}{18\!\cdots\!25}a^{7}+\frac{18\!\cdots\!23}{73\!\cdots\!00}a^{6}-\frac{12\!\cdots\!69}{36\!\cdots\!50}a^{5}+\frac{23\!\cdots\!51}{21\!\cdots\!00}a^{4}+\frac{12\!\cdots\!33}{26\!\cdots\!75}a^{3}+\frac{15\!\cdots\!63}{73\!\cdots\!00}a^{2}+\frac{35\!\cdots\!17}{14\!\cdots\!00}a-\frac{29\!\cdots\!67}{73\!\cdots\!00}$, $\frac{1}{16\!\cdots\!00}a^{44}-\frac{60\!\cdots\!61}{16\!\cdots\!00}a^{43}-\frac{21\!\cdots\!67}{16\!\cdots\!00}a^{42}+\frac{46\!\cdots\!83}{16\!\cdots\!00}a^{41}-\frac{19\!\cdots\!27}{16\!\cdots\!00}a^{40}+\frac{38\!\cdots\!39}{16\!\cdots\!00}a^{39}+\frac{13\!\cdots\!96}{59\!\cdots\!75}a^{38}+\frac{24\!\cdots\!76}{83\!\cdots\!25}a^{37}-\frac{21\!\cdots\!57}{83\!\cdots\!50}a^{36}+\frac{19\!\cdots\!79}{83\!\cdots\!50}a^{35}-\frac{10\!\cdots\!13}{16\!\cdots\!00}a^{34}+\frac{29\!\cdots\!83}{34\!\cdots\!00}a^{33}-\frac{44\!\cdots\!56}{41\!\cdots\!25}a^{32}-\frac{10\!\cdots\!64}{41\!\cdots\!25}a^{31}+\frac{59\!\cdots\!42}{41\!\cdots\!25}a^{30}+\frac{12\!\cdots\!27}{83\!\cdots\!50}a^{29}+\frac{11\!\cdots\!69}{41\!\cdots\!25}a^{28}-\frac{49\!\cdots\!46}{16\!\cdots\!65}a^{27}+\frac{29\!\cdots\!43}{41\!\cdots\!25}a^{26}-\frac{25\!\cdots\!79}{83\!\cdots\!50}a^{25}-\frac{24\!\cdots\!17}{11\!\cdots\!75}a^{24}+\frac{34\!\cdots\!13}{12\!\cdots\!75}a^{23}-\frac{12\!\cdots\!84}{41\!\cdots\!25}a^{22}+\frac{25\!\cdots\!68}{41\!\cdots\!25}a^{21}+\frac{14\!\cdots\!63}{16\!\cdots\!00}a^{20}-\frac{18\!\cdots\!59}{16\!\cdots\!00}a^{19}+\frac{60\!\cdots\!17}{16\!\cdots\!00}a^{18}+\frac{64\!\cdots\!77}{16\!\cdots\!00}a^{17}+\frac{72\!\cdots\!89}{16\!\cdots\!00}a^{16}+\frac{24\!\cdots\!37}{16\!\cdots\!00}a^{15}+\frac{72\!\cdots\!23}{16\!\cdots\!00}a^{14}-\frac{88\!\cdots\!87}{95\!\cdots\!80}a^{13}+\frac{50\!\cdots\!07}{16\!\cdots\!00}a^{12}+\frac{11\!\cdots\!59}{22\!\cdots\!00}a^{11}+\frac{73\!\cdots\!21}{17\!\cdots\!50}a^{10}-\frac{23\!\cdots\!68}{41\!\cdots\!25}a^{9}+\frac{23\!\cdots\!99}{83\!\cdots\!50}a^{8}+\frac{55\!\cdots\!41}{11\!\cdots\!50}a^{7}+\frac{11\!\cdots\!69}{41\!\cdots\!25}a^{6}-\frac{39\!\cdots\!79}{83\!\cdots\!50}a^{5}-\frac{53\!\cdots\!99}{11\!\cdots\!50}a^{4}-\frac{20\!\cdots\!86}{41\!\cdots\!25}a^{3}+\frac{51\!\cdots\!79}{16\!\cdots\!00}a^{2}+\frac{64\!\cdots\!17}{16\!\cdots\!00}a+\frac{59\!\cdots\!12}{93\!\cdots\!25}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, 1/2*a^20 - 1/2*a^18 - 1/2*a^17 - 1/2*a^16 - 1/2*a^8 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/14*a^21 - 1/2*a^19 - 1/2*a^18 - 1/2*a^17 + 2/7*a^15 + 3/14*a^9 + 5/14*a^7 - 1/2*a^5 - 1/2*a^4 - 1/14*a^3 - 1/2*a^2 - 5/14*a + 1/7, 1/14*a^22 - 1/2*a^19 - 1/2*a^17 - 3/14*a^16 + 3/14*a^10 - 1/7*a^8 - 1/2*a^5 + 3/7*a^4 + 1/7*a^2 - 5/14*a - 1/2, 1/14*a^23 + 2/7*a^17 - 1/2*a^16 + 3/14*a^11 - 1/7*a^9 - 1/2*a^8 + 3/7*a^5 - 1/2*a^4 - 5/14*a^3 + 1/7*a^2 - 1/2, 1/14*a^24 + 2/7*a^18 - 1/2*a^17 + 3/14*a^12 - 1/7*a^10 - 1/2*a^9 + 3/7*a^6 - 1/2*a^5 - 5/14*a^4 + 1/7*a^3 - 1/2*a, 1/14*a^25 + 2/7*a^19 - 1/2*a^18 + 3/14*a^13 - 1/7*a^11 - 1/2*a^10 + 3/7*a^7 - 1/2*a^6 - 5/14*a^5 + 1/7*a^4 - 1/2*a^2, 1/14*a^26 - 3/14*a^20 - 1/2*a^19 - 1/2*a^18 - 1/2*a^17 - 1/2*a^16 + 3/14*a^14 - 1/7*a^12 - 1/2*a^11 - 1/14*a^8 - 1/2*a^7 + 1/7*a^6 + 1/7*a^5 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/14*a^27 - 1/2*a^18 - 1/2*a^17 - 1/2*a^16 + 1/14*a^15 - 1/7*a^13 - 1/2*a^12 - 3/7*a^9 + 3/14*a^7 - 5/14*a^6 - 3/14*a^3 - 1/2*a^2 - 1/14*a - 1/14, 1/14*a^28 - 1/2*a^19 - 1/2*a^18 - 1/2*a^17 + 1/14*a^16 - 1/7*a^14 - 1/2*a^13 - 3/7*a^10 + 3/14*a^8 - 5/14*a^7 - 3/14*a^4 - 1/2*a^3 - 1/14*a^2 - 1/14*a, 1/14*a^29 - 1/2*a^19 - 3/7*a^17 - 1/2*a^16 - 1/7*a^15 - 1/2*a^14 - 3/7*a^11 + 3/14*a^9 + 1/7*a^8 - 1/2*a^6 - 3/14*a^5 + 3/7*a^3 + 3/7*a^2 - 1/2*a - 1/2, 1/350*a^30 - 1/35*a^29 + 1/70*a^28 - 1/35*a^27 + 1/35*a^26 + 6/175*a^25 + 1/35*a^23 - 1/70*a^22 + 4/35*a^20 + 16/35*a^19 - 2/35*a^18 + 3/35*a^17 + 33/70*a^16 - 67/350*a^15 + 16/35*a^14 - 1/10*a^13 - 11/35*a^12 - 13/35*a^11 + 11/25*a^10 - 17/35*a^9 + 17/35*a^8 + 8/35*a^7 - 1/35*a^6 - 111/350*a^5 + 9/35*a^4 - 13/35*a^3 + 17/70*a^2 + 2/7*a - 16/175, 1/350*a^31 + 1/70*a^29 - 1/35*a^28 + 1/35*a^27 + 6/175*a^26 - 1/70*a^25 + 1/35*a^24 - 1/70*a^23 - 1/35*a^21 - 3/70*a^20 + 3/35*a^19 - 17/35*a^18 - 1/35*a^17 + 79/175*a^16 - 11/35*a^15 - 1/10*a^14 + 3/70*a^13 + 2/35*a^12 - 23/175*a^11 - 3/10*a^10 - 3/35*a^9 - 19/70*a^8 - 1/35*a^7 + 139/350*a^6 - 19/70*a^5 + 19/70*a^4 + 16/35*a^3 + 2/7*a^2 + 43/350*a + 1/70, 1/350*a^32 - 1/35*a^29 + 1/35*a^28 + 6/175*a^27 - 1/70*a^26 - 1/70*a^24 - 1/35*a^22 + 1/35*a^21 + 3/35*a^20 + 3/10*a^19 + 9/35*a^18 - 17/350*a^17 - 27/70*a^16 + 2/7*a^15 + 3/70*a^14 - 8/35*a^13 + 27/175*a^12 - 31/70*a^11 + 1/14*a^10 - 17/35*a^9 + 33/70*a^8 - 111/350*a^7 - 9/70*a^6 - 2/7*a^5 + 11/35*a^4 + 3/7*a^3 - 66/175*a^2 - 12/35*a + 17/70, 1/2450*a^33 + 1/1225*a^32 + 3/2450*a^31 - 1/35*a^29 + 1/1225*a^28 + 1/50*a^27 + 13/1225*a^26 + 1/98*a^25 - 11/490*a^24 + 1/49*a^23 - 1/70*a^22 + 9/490*a^21 + 12/245*a^20 + 93/490*a^19 + 79/350*a^18 + 538/1225*a^17 + 79/2450*a^16 + 153/490*a^15 - 68/245*a^14 - 418/1225*a^13 + 319/1225*a^12 + 151/1225*a^11 + 11/490*a^10 + 26/245*a^9 + 709/2450*a^8 - 73/175*a^7 + 27/2450*a^6 - 31/245*a^5 - 72/245*a^4 - 76/1225*a^3 - 81/175*a^2 - 13/1225*a + 69/245, 1/2450*a^34 - 1/2450*a^32 + 1/2450*a^31 + 1/1225*a^29 - 1/98*a^28 - 1/1225*a^27 + 57/2450*a^26 + 11/490*a^24 + 1/490*a^23 - 6/245*a^22 - 4/245*a^21 + 12/245*a^20 + 883/2450*a^19 - 87/245*a^18 - 393/2450*a^17 + 69/1225*a^16 - 67/245*a^15 - 473/1225*a^14 - 31/70*a^13 - 309/2450*a^12 - 523/1225*a^11 + 92/245*a^10 - 3/350*a^9 + 57/245*a^8 - 99/2450*a^7 + 6/175*a^6 + 109/245*a^5 - 171/350*a^4 - 117/490*a^3 - 279/1225*a^2 - 38/175*a - 61/245, 1/2450*a^35 + 3/2450*a^32 + 3/2450*a^31 + 1/1225*a^30 + 8/245*a^29 - 69/2450*a^27 + 13/1225*a^26 + 8/245*a^25 - 1/49*a^24 - 1/245*a^23 - 3/98*a^22 - 1/245*a^21 - 111/1225*a^20 - 81/490*a^19 - 213/490*a^18 - 1061/2450*a^17 + 317/1225*a^16 + 522/1225*a^15 - 54/245*a^14 - 159/490*a^13 + 817/2450*a^12 + 86/1225*a^11 + 17/1225*a^10 - 57/245*a^9 - 53/490*a^8 + 8/175*a^7 - 229/1225*a^6 - 807/2450*a^5 + 229/490*a^4 - 37/490*a^3 + 87/350*a^2 + 207/1225*a + 103/490, 1/12250*a^36 + 1/6125*a^35 - 1/12250*a^34 - 1/12250*a^33 + 1/6125*a^32 + 8/6125*a^31 - 1/1750*a^30 - 61/6125*a^29 + 333/12250*a^28 - 61/12250*a^27 + 13/12250*a^26 + 194/6125*a^25 + 53/2450*a^24 + 17/2450*a^23 - 3/1225*a^22 - 417/12250*a^21 - 186/875*a^20 - 1223/12250*a^19 - 1304/6125*a^18 + 1518/6125*a^17 + 68/6125*a^16 + 347/1225*a^15 - 4929/12250*a^14 - 5969/12250*a^13 + 2459/6125*a^12 - 292/1225*a^11 + 2439/12250*a^10 + 433/6125*a^9 + 3561/12250*a^8 - 1087/12250*a^7 - 2111/12250*a^6 - 5163/12250*a^5 - 2194/6125*a^4 + 2536/6125*a^3 + 1683/6125*a^2 + 1966/6125*a - 302/875, 1/12250*a^37 + 1/12250*a^34 - 1/12250*a^33 + 17/12250*a^32 - 2/6125*a^31 + 1/1750*a^30 - 423/12250*a^29 + 313/12250*a^28 - 4/175*a^27 + 41/6125*a^26 - 13/6125*a^25 - 1/175*a^24 + 3/98*a^23 - 207/12250*a^22 + 23/1225*a^21 - 3/35*a^20 - 1206/6125*a^19 + 2181/6125*a^18 + 3679/12250*a^17 + 1289/6125*a^16 + 2008/6125*a^15 - 5211/12250*a^14 + 3411/12250*a^13 - 418/6125*a^12 - 631/12250*a^11 - 186/6125*a^10 + 3279/12250*a^9 + 2321/12250*a^8 - 328/875*a^7 + 2412/6125*a^6 + 4223/12250*a^5 - 3026/6125*a^4 + 2566/6125*a^3 - 9/25*a^2 + 619/6125*a - 71/250, 1/12250*a^38 + 1/12250*a^35 - 1/12250*a^34 + 1/6125*a^33 + 1/12250*a^32 - 3/12250*a^31 - 3/12250*a^30 - 387/12250*a^29 + 4/1225*a^28 + 146/6125*a^27 - 173/6125*a^26 + 9/2450*a^25 - 3/98*a^24 - 216/6125*a^23 + 11/2450*a^22 + 1/490*a^21 + 869/6125*a^20 + 1406/6125*a^19 - 4441/12250*a^18 + 5093/12250*a^17 - 1439/12250*a^16 - 2301/12250*a^15 + 3111/12250*a^14 - 27/125*a^13 + 2407/6125*a^12 - 2206/6125*a^11 + 3959/12250*a^10 - 177/6125*a^9 - 876/6125*a^8 - 1231/12250*a^7 - 1467/12250*a^6 + 2903/12250*a^5 - 212/875*a^4 + 1149/2450*a^3 - 547/12250*a^2 + 3841/12250*a + 597/2450, 1/1849750*a^39 + 3/924875*a^38 + 71/1849750*a^37 - 71/1849750*a^36 + 181/1849750*a^35 - 163/924875*a^34 + 157/924875*a^33 - 549/1849750*a^32 - 1847/1849750*a^31 - 1599/1849750*a^30 + 59589/1849750*a^29 - 1614/132125*a^28 - 30087/1849750*a^27 - 817/36995*a^26 + 7999/264250*a^25 + 18584/924875*a^24 - 27857/1849750*a^23 + 40543/1849750*a^22 + 421/924875*a^21 + 201069/924875*a^20 + 50644/184975*a^19 - 5295/14798*a^18 - 689809/1849750*a^17 + 884441/1849750*a^16 + 41093/264250*a^15 - 685333/1849750*a^14 + 15133/132125*a^13 - 112673/369950*a^12 + 18759/37750*a^11 + 4056/26425*a^10 + 12714/132125*a^9 + 246203/924875*a^8 + 412497/924875*a^7 + 159056/924875*a^6 - 433218/924875*a^5 - 775939/1849750*a^4 - 333482/924875*a^3 + 35001/924875*a^2 + 177511/369950*a + 50331/924875, 1/606040991500*a^40 + 9091/60604099150*a^39 + 920781/151510247875*a^38 + 1302657/151510247875*a^37 + 13478903/606040991500*a^36 + 24428647/303020495750*a^35 - 8413093/121208198300*a^34 - 58495473/303020495750*a^33 - 818509317/606040991500*a^32 + 151275571/303020495750*a^31 - 33119853/303020495750*a^30 + 40844813/1731545690*a^29 - 1013678762/151510247875*a^28 - 528051212/21644321125*a^27 - 4158481912/151510247875*a^26 + 6521589719/303020495750*a^25 - 372340641/30302049575*a^24 + 6327643111/303020495750*a^23 + 286405683/21644321125*a^22 + 1970641077/303020495750*a^21 - 6795016999/60604099150*a^20 - 21588964/123681835*a^19 - 22494657309/151510247875*a^18 - 4833311913/43288642250*a^17 - 55573513873/121208198300*a^16 + 45787456098/151510247875*a^15 - 12133922328/30302049575*a^14 - 9613254138/30302049575*a^13 - 10139366183/24241639660*a^12 - 8931025276/151510247875*a^11 + 113290147073/303020495750*a^10 - 18607240299/60604099150*a^9 + 198714302831/606040991500*a^8 + 96043037791/303020495750*a^7 - 5478157681/86577284500*a^6 + 18940543149/151510247875*a^5 - 8298452499/17315456900*a^4 - 34327164191/303020495750*a^3 - 133549729239/606040991500*a^2 - 32422606813/151510247875*a - 47998872399/121208198300, 1/606040991500*a^41 + 187/6060409915*a^39 - 72354/21644321125*a^38 - 719567/121208198300*a^37 - 133963/12120819830*a^36 + 23381189/606040991500*a^35 + 8393647/151510247875*a^34 + 1325411/86577284500*a^33 + 89318109/303020495750*a^32 + 50468422/151510247875*a^31 - 22359226/21644321125*a^30 + 295915964/151510247875*a^29 + 3057727341/303020495750*a^28 - 3557321426/151510247875*a^27 + 6881114131/303020495750*a^26 + 382192369/303020495750*a^25 - 11546187/865772845*a^24 + 160689703/21644321125*a^23 - 951096079/30302049575*a^22 - 1978765913/151510247875*a^21 - 10437793077/43288642250*a^20 + 53296929191/151510247875*a^19 + 3005873588/6060409915*a^18 + 45701950319/606040991500*a^17 - 135232171149/303020495750*a^16 - 44280446611/303020495750*a^15 + 75394032423/151510247875*a^14 + 42220214143/86577284500*a^13 + 35285183351/303020495750*a^12 - 4628171819/21644321125*a^11 + 110006041681/303020495750*a^10 - 5951837299/86577284500*a^9 - 51795066251/303020495750*a^8 - 141730455163/606040991500*a^7 + 8656466703/43288642250*a^6 - 24876124257/606040991500*a^5 + 3485296456/151510247875*a^4 - 109316218647/606040991500*a^3 + 12079059186/151510247875*a^2 + 29704633201/606040991500*a + 63844060038/151510247875, 1/73850355247017487558330112796800194219766500*a^42 - 50775037638297295761106693905757/73850355247017487558330112796800194219766500*a^41 + 28083398287283825213525623595283/36925177623508743779165056398400097109883250*a^40 + 195318103210795148224519644412363391/2637512687393481698511789742742864079277375*a^39 - 1150185429195729692948826595441977832059/73850355247017487558330112796800194219766500*a^38 + 2888072507200942501006197629629743767869/73850355247017487558330112796800194219766500*a^37 - 1628289726171755212504069741972239278993/73850355247017487558330112796800194219766500*a^36 + 5416598774176403091300619420455572413517/73850355247017487558330112796800194219766500*a^35 - 45250690898643432578833190840797456611/73850355247017487558330112796800194219766500*a^34 - 13253159362457792310775121579587722134117/73850355247017487558330112796800194219766500*a^33 - 2425509010269494186531430850458417652334/3692517762350874377916505639840009710988325*a^32 - 370069149021809078411886748176150084271/1055005074957392679404715897097145631710950*a^31 - 8487054905769665705017203533531998505573/36925177623508743779165056398400097109883250*a^30 - 72579542699691737389921154398999463988043/2637512687393481698511789742742864079277375*a^29 - 221405986596937730613773945747582573337997/18462588811754371889582528199200048554941625*a^28 + 29723271446192191226583671065201095838224/3692517762350874377916505639840009710988325*a^27 + 9549045838977435210386653374923834373769/3692517762350874377916505639840009710988325*a^26 - 73497029651858456810459100253311082048851/18462588811754371889582528199200048554941625*a^25 + 398658212137259193781641324198001611339911/18462588811754371889582528199200048554941625*a^24 - 241476759603775104668547655502609634025968/18462588811754371889582528199200048554941625*a^23 - 1064181995881090997727217047845176241057/1477007104940349751166602255936003884395330*a^22 + 10647848362805439269009413773346144990746/527502537478696339702357948548572815855475*a^21 - 585564133720817499302730477963701244360639/18462588811754371889582528199200048554941625*a^20 + 4574559801181398895227986129451804038385859/18462588811754371889582528199200048554941625*a^19 + 9127472233370629271953115857648850193579227/73850355247017487558330112796800194219766500*a^18 + 3469444568184745570796977505454598707914779/14770071049403497511666022559360038843953300*a^17 - 479998012259433505985158084688483721056648/3692517762350874377916505639840009710988325*a^16 + 730999906987651386172477416713642783404854/18462588811754371889582528199200048554941625*a^15 + 3885201175181739504155958027128993507274957/10550050749573926794047158970971456317109500*a^14 - 3933485134306335001395724970550496447601997/73850355247017487558330112796800194219766500*a^13 + 206301134413188253560554675494694102725667/738503552470174875583301127968001942197665*a^12 - 166973226752826800580636557811692659324467/1055005074957392679404715897097145631710950*a^11 + 17034152956798041001018063367137216841766859/73850355247017487558330112796800194219766500*a^10 - 9035743413520731272833103125332576930023619/73850355247017487558330112796800194219766500*a^9 - 21756768636740764786653396716394101403670373/73850355247017487558330112796800194219766500*a^8 + 141027497507020481109280131969559515521869/422002029982957071761886358838858252684380*a^7 + 3737476742869353183888552749687799729060027/14770071049403497511666022559360038843953300*a^6 - 36150364838113700370160020821221146608361859/73850355247017487558330112796800194219766500*a^5 - 226562585272416342953111173707269576353099/1507150107081989542006736995853065188158500*a^4 - 322787048136463067975617766408516054680253/1507150107081989542006736995853065188158500*a^3 + 5150140517234577528875404491983621164415289/10550050749573926794047158970971456317109500*a^2 - 5064565093631289959789977239447702953510251/73850355247017487558330112796800194219766500*a - 13251285987844581798470935058745472643955039/36925177623508743779165056398400097109883250, 1/73850355247017487558330112796800194219766500*a^43 + 60724143780869973977604868617741/73850355247017487558330112796800194219766500*a^41 + 5201562368565622438818202862981/14770071049403497511666022559360038843953300*a^40 - 18479509677585345276247346289165151829/73850355247017487558330112796800194219766500*a^39 + 615425989403081477018563886466547114139/36925177623508743779165056398400097109883250*a^38 - 46117936963612197294132635780744558909/7385035524701748755833011279680019421976650*a^37 + 1749778608040513566883584494227491293663/73850355247017487558330112796800194219766500*a^36 - 503090264644875509767711546563005408197/36925177623508743779165056398400097109883250*a^35 + 13917188149398103849048441260772713487513/73850355247017487558330112796800194219766500*a^34 - 1051779764555382094631721699245065738389/14770071049403497511666022559360038843953300*a^33 - 94075827396956530022633196439216995783939/73850355247017487558330112796800194219766500*a^32 + 31596160730048176030803561427719123985481/36925177623508743779165056398400097109883250*a^31 - 20618082951651798091257566257064490354868/18462588811754371889582528199200048554941625*a^30 + 1261699063583302776566356124208756650390721/36925177623508743779165056398400097109883250*a^29 + 410822927988400073428321712373074319025797/36925177623508743779165056398400097109883250*a^28 - 620048024557675532138933965044146634996127/18462588811754371889582528199200048554941625*a^27 - 735452659070869180859609310333579513190711/36925177623508743779165056398400097109883250*a^26 - 1085625780533739849059828391849275183504219/36925177623508743779165056398400097109883250*a^25 + 372381518328818630591714912770432054356599/36925177623508743779165056398400097109883250*a^24 - 454655756664422082384635177150177175297252/18462588811754371889582528199200048554941625*a^23 + 58060029095612266401109897690612186253988/3692517762350874377916505639840009710988325*a^22 - 101770613709889703635706463760585348012132/18462588811754371889582528199200048554941625*a^21 - 3204947896271147175386286675754442245891838/18462588811754371889582528199200048554941625*a^20 + 6991933329248163408783664858680320586529447/73850355247017487558330112796800194219766500*a^19 + 8703105450152485117176853207376534956231063/18462588811754371889582528199200048554941625*a^18 + 3025348276177221497196070231453529588245309/10550050749573926794047158970971456317109500*a^17 - 5420728558454455668161471250718735589650407/73850355247017487558330112796800194219766500*a^16 - 34524406582957740421786619740663445811974113/73850355247017487558330112796800194219766500*a^15 + 2650126023616114261774583418502180639775627/18462588811754371889582528199200048554941625*a^14 + 33724560170559831701775528495394697040733249/73850355247017487558330112796800194219766500*a^13 - 4944953650559609573253487360176107902989191/73850355247017487558330112796800194219766500*a^12 - 20470895861754034273379783485754427094400273/73850355247017487558330112796800194219766500*a^11 + 17567981282109246102533787560372057962571309/36925177623508743779165056398400097109883250*a^10 + 393802379742144353357111226608044422912704/2637512687393481698511789742742864079277375*a^9 + 27409523905083986509108398109682599678769/14770071049403497511666022559360038843953300*a^8 + 6752253898201094308044464904814463672682526/18462588811754371889582528199200048554941625*a^7 + 18195607808784970044141328250574749076950623/73850355247017487558330112796800194219766500*a^6 - 12717110319737009120836928242443568761576369/36925177623508743779165056398400097109883250*a^5 + 23351914582683402678932184869903987873651/215307158154569934572390999407580741165500*a^4 + 1210605206037696103208703698934930240061233/2637512687393481698511789742742864079277375*a^3 + 15327203837206062073947764148300990762675863/73850355247017487558330112796800194219766500*a^2 + 3559253046676629013157544713340742171475017/14770071049403497511666022559360038843953300*a - 29212158738188194648080702599037662553562667/73850355247017487558330112796800194219766500, 1/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^44 - 60311681676044293023017683096476547363087730046918516970559816078626104768926166218283955692825138822911379300309488716161028939135361305942367961/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^43 - 21441957629297790939402224415036900334064153618826474555044263831441134127164203515757986660203503002320706218979989326261691590472686819353839967/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^42 + 467724346673775575012144336648203979133385627150010616042520698732870986398177902807065381208788359348592001898260193329317683536486574031210836446043285997926229853226975757683/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^41 - 1909573890833443617402637091601509289452382925217360462669688220585899876117343099612315335473693797916934141183873223484347947172470324097923302594132545024517376129434172904527/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^40 + 3844705926420171289261816595360275140872618823966434166352056136204302074692076389412833923151473738190881499007753902530616984664565202050613884002568382263478422533265651140469326439/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^39 + 13356912843448899098785147974892723163925828350130150709441747473379017801244219048250894507924186069677432204678989279701763672495706444872751648095607128142191755185764949928511633796/597829658738029000103466366465564824973428694212769807719537574824237772460046749377317431081682872297031276746500242965680563080471436005727004211055603214232204644571662014380250783527375*a^38 + 24473573802257684778598132113625929820325052794174006199667372494130361874800863631802475773694363452365303999119919778584055882868610457469997001108604724912413644847200812102070418076/836961522233240600144852913051790754962800171897877730807352604753932881444065449128244403514356021215843787445100340151952788312660010408017805895477844499925086502400326820132351096938325*a^37 - 21044302257257778157424189113294839827968156013928492919327466918771876601997098066894242410928915785396609363617412073930865189246213853310693718523017823939182810516865512334797873257/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^36 + 195479801857370345434942232517791415327676219595919514586225994093312772000437652565100523229780720546629311820706282108547931266811631226610486547117568851072483664379695376416058604279/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^35 - 1090025736776330806931504267191347197220704676328474570332038190540782050046373669726026456482090365085081554056374409585164854918469035452280276920506888251048358766355887415341827083713/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^34 + 29123339399418285843585148217609738979061771013881038791203062729627526345869204860953501837292024283065606817078166907710290286597403312818187957191513222800332774558986776451718594383/341616947850302285773409352266037042841959253835868461554021471328135869977169571072752817760961641312589300998000138837531750331697963431844002406317487550989831225469521151074429019158500*a^33 - 4405535851480180275573221172681258534987112585026368874353474995798274502733562527429688824426821993720299627083483126067421569720323450835069225266030457494233210463857246105835550180656/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^32 - 1079579733232527850151993160489543922766462307850835902992536915552451907017176826671696736100417790893384170128662084770852017405428060161754941129160646206291026198622227559375162217464/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^31 + 5956194875509359371827380299550726851174868785778085653951832935842474166621661910014513978082796110959479318551484747300189639367693457825635771413911062142282472128808402354217010018342/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^30 + 127019404555483637197562855514585646814100289062691006423643853609188604648459302793630020022863775669911195379929536310939345165422752038164192957442444896834007978419316719120582262867827/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^29 + 117834580352191153308007439178091411519082292790779320716747413116846441969937038389938774230271376104200535094320657170128836096546386042734272414175318937480860625366522851749713612709169/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^28 - 4950421367637807389234501296298749856095171966845424229308627007929913714547785578319085798789318748293009276359406040051003747113217078995331615560275023577534956624370760186438867572046/167392304446648120028970582610358150992560034379575546161470520950786576288813089825648880702871204243168757489020068030390557662532002081603561179095568899985017300480065364026470219387665*a^27 + 29024720154365459154773331253918440091734479512502749297205750660153629338713326341400245477640747938740527004374973159600894700564400106504075610755285804202571216422817315833975221327843/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^26 - 254989591199554493822870459114457834601926735124511482703227505833510152751194040925195516646461351254689378630005034271858252023376204912818909506899241274616454674497238147857946078554179/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^25 - 241370746002261460662864138410301214911328468762618807888863393057990245323647252099076102545692181796702940721002730178915667525142819494531696364907984843592807319559541999345561648817/119565931747605800020693273293112964994685738842553961543907514964847554492009349875463486216336574459406255349300048593136112616094287201145400842211120642846440928914332402876050156705475*a^24 + 340449673724170845876701061706959114243411978065666437416401240093926280913534349538710787191959232457729540114273445593534755778109791835233714519539303354412959582292654681571490948213/12200605280367938777621762580929894387212830494138159341215052547433423927756056109741172062891487189735332178500004958483276797560641551137285800225624555392493972338197183966943893541375*a^23 - 1220270355311984185967176190732934682112138343539615423032559347551488596220560955570268853797540400688259084069582343405344714844523176153611050254305162781008325818104386954255583032284/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^22 + 2538729457193345592055841188597820313565376370994665085300584887030632256966625534035673098946725945735085521190889732498289623156676845883133351691526740719313959468130716740306224689668/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^21 + 1461652999473836731068606782806308673289607882077933108137165236193950985017464247378474000203947946385924653367507279465597640654036083969318930812032014841216758175574060333764657038823663/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^20 - 1899804001565736960458969643178895106656763460517465932781164881662116198802942001168468669453479308032906852491513903900645080518471687188568838113352794631162109131073765747115837817551359/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^19 + 6029470470398858820962646285020790611737092873777623848105377226969054035752020762952803211426597626144714043564597319114718991222778258700271514621095704747920381008219217310121410685154917/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^18 + 6463873966569457434316552762156400775165112175423352279626633139431442990195457447725728900976329452416332188943168568776086535737049168983232861348193072870641083620163910974620976061734477/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^17 + 7242189364364290045192477195378437709257175786363339342655375566856297268939841692159122387597777477153009626960850025124728005624912267856870595392773903814467433073808564635401235515300889/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^16 + 2495094816260224823300755468013208316906222935274541181357497699640646295767591368771872560519651725410615977001140619816335919118949755832016762949017627093595902442794670360444214396993237/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^15 + 721077141571227828499065954108530732740989834507242256213687094626656370446139428371372021327400848449910252690298299397065164900915164732932303195425182059096388122226435543054607773017923/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^14 - 8891925304200829699125374092686092723960512688400127292656272397284586802610959864991602098120629749577635095257709385508428691365840068379155168323778957534895806539120722536869664861387/95652745398084640016554618634490371995748591074043169235126011971878043593607479900370788973069259567525004279440038874508890092875429760916320673768896514277152743131465922300840125364380*a^13 + 5059878771202297186886862148540607981878817943863354015982598299025611114583690992138915291763880944964955759550622185970227471150537622430584905738909746522522885714316099089474042618885907/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^12 + 11015671397895997772520913471460582106081961714579173922486368828595321373387387409170783188371661396278782945549969940996259273307214206711381598588681756570808829228260621208826286424959/22171166151873923182645110279517635893054309189347754458472916682223387587922263553066076914287576720949504303181465964290140087752583057166034593257691245031128119268882829672380161508300*a^11 + 73462523386356674571661431539363680263852057879374795025276457261243550038693078839652023022856442924748433719435710880332321742574486588857735910896187928194914242657122606115095105390521/170808473925151142886704676133018521420979626917934230777010735664067934988584785536376408880480820656294650499000069418765875165848981715922001203158743775494915612734760575537214509579250*a^10 - 234379283779384149812794093516491191257289204086636348597444358783617396093506094568118538084031802765636425100534929198664841705958029394767100686865854089957157237934600563656880507347668/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^9 + 2364206258949628808174123875882078635056806762489873691126161061776978703005935166523074648977647901161752223598287158344132189468891348998631012206333684752160884187100095782831939987212899/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^8 + 55396049700181599617635707365403640423109104308197165058055124665562689174332607781013842987806557896121843500280374980851022114425407892227671638393361575771200790686605985410517118202541/1195659317476058000206932732931129649946857388425539615439075149648475544920093498754634862163365744594062553493000485931361126160942872011454008422111206428464409289143324028760501567054750*a^7 + 1155033821408827888915804009174530608059434441886342008593571653638584547335844274147803565823363219255425503972339790342671960586474771103484479796289017166009090583941119326365163780351369/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^6 - 3958857320024038968693668962296860169185720282852860980217520797475631610372319615945239428674821850090418631676613540826601845128029349681580501172451431506733651850754699986880342489266579/8369615222332406001448529130517907549628001718978777308073526047539328814440654491282444035143560212158437874451003401519527883126600104080178058954778444999250865024003268201323510969383250*a^5 - 538116137048534397454333146720694856109302843364007825577020814362926902001110521858654553484602720534107242446785391437070963795968936435497191296739577921592039482971166256271489388514599/1195659317476058000206932732931129649946857388425539615439075149648475544920093498754634862163365744594062553493000485931361126160942872011454008422111206428464409289143324028760501567054750*a^4 - 2071292097798335902703681396711449583369501583103157285395110230381494783321919096352038941449838657964839332874729387131918722625136258341421959792765679825369774519879240256721272190403586/4184807611166203000724264565258953774814000859489388654036763023769664407220327245641222017571780106079218937225501700759763941563300052040089029477389222499625432512001634100661755484691625*a^3 + 5119681135927267626172087843186337190065392734553841123833640278993888468740188921185992223702918936685068258395702116279880002450933925705191426362028775547852810158512839181676660739726979/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a^2 + 6432085122479697657658423413277984388668368359442972167036309210503442449239874258670540978316090917428317486703764234028213299494889092977657993279087851968306666319013144902553183855502917/16739230444664812002897058261035815099256003437957554616147052095078657628881308982564888070287120424316875748902006803039055766253200208160356117909556889998501730048006536402647021938766500*a + 591738035562266722799947703703474944483223258379731921747887509736787316518551856238675759364088316618569922286450367611697923500052538813120563723889524744638699378110574193990009304412/9320284211951454344597471192113482794685970733829373394291231678774308256615428164011630328667661706189797187584636304587447531321380962227369776118906954342150183768377804233099678139625

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$44$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	not computed	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 15*x^44 - 120*x^43 + 2785*x^42 + 2040*x^41 - 222342*x^40 + 420425*x^39 + 10024590*x^38 - 35069730*x^37 - 280556895*x^36 + 1389459441*x^35 + 4980489270*x^34 - 34327919505*x^33 - 52485204525*x^32 + 576080544270*x^31 + 196092367854*x^30 - 6820772763030*x^29 + 3051952728270*x^28 + 57922300812880*x^27 - 59937846525180*x^26 - 353495076875202*x^25 + 542694665981220*x^24 + 1531071286376460*x^23 - 3131712122030760*x^22 - 4545064586341055*x^21 + 12339167509476885*x^20 + 8461213880546430*x^19 - 33709107574234355*x^18 - 6968642817313350*x^17 + 63373889569191960*x^16 - 7076452922900098*x^15 - 80218608988132905*x^14 + 26470314281644410*x^13 + 66243722166245240*x^12 - 32060240741785995*x^11 - 34356066621763764*x^10 + 20781515138515650*x^9 + 10477766738441505*x^8 - 7675031336820660*x^7 - 1594722107742745*x^6 + 1559444254090284*x^5 + 49643486026125*x^4 - 150881459933485*x^3 + 10931860799430*x^2 + 4297070637735*x - 401780151251);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_3\times C_{15}$ (as 45T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

An abelian group of order 45

The 45 conjugacy class representatives for $C_3\times C_{15}$

Character table for $C_3\times C_{15}$

Intermediate fields

$\Q(\zeta_{9})^+$, 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 5.5.390625.1, 9.9.25002110044521.1, 15.15.207828545629978179931640625.1, 15.15.365440026390612125396728515625.1, 15.15.1274210583519814691674768924713134765625.1, 15.15.1274210583519814691674768924713134765625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$15^{3}$	R	R	${\href{/padicField/7.3.0.1}{3} }^{15}$	$15^{3}$	$15^{3}$	$15^{3}$	R	$15^{3}$	$15^{3}$	$15^{3}$	${\href{/padicField/37.5.0.1}{5} }^{9}$	$15^{3}$	${\href{/padicField/43.3.0.1}{3} }^{15}$	$15^{3}$	$15^{3}$	$15^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3		Deg $45$	$3$	$15$	$60$
$5$ 5	5.15.24.88	$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$	$5$	$3$	$24$	$C_{15}$	$[2]^{3}$
	5.15.24.88	$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$	$5$	$3$	$24$	$C_{15}$	$[2]^{3}$
	5.15.24.88	$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$	$5$	$3$	$24$	$C_{15}$	$[2]^{3}$
$19$ 19	19.15.10.1	$x^{15} + 95 x^{12} + 15 x^{11} + 51 x^{10} + 3610 x^{9} - 4275 x^{8} - 28995 x^{7} + 69100 x^{6} - 31623 x^{5} + 835620 x^{4} + 700180 x^{3} + 1194855 x^{2} - 1636410 x + 2391332$	$3$	$5$	$10$	$C_{15}$	$[\ ]_{3}^{5}$
	19.15.10.1	$x^{15} + 95 x^{12} + 15 x^{11} + 51 x^{10} + 3610 x^{9} - 4275 x^{8} - 28995 x^{7} + 69100 x^{6} - 31623 x^{5} + 835620 x^{4} + 700180 x^{3} + 1194855 x^{2} - 1636410 x + 2391332$	$3$	$5$	$10$	$C_{15}$	$[\ ]_{3}^{5}$
	19.15.10.1	$x^{15} + 95 x^{12} + 15 x^{11} + 51 x^{10} + 3610 x^{9} - 4275 x^{8} - 28995 x^{7} + 69100 x^{6} - 31623 x^{5} + 835620 x^{4} + 700180 x^{3} + 1194855 x^{2} - 1636410 x + 2391332$	$3$	$5$	$10$	$C_{15}$	$[\ ]_{3}^{5}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more