LMFDB - Number field 45.45.13198639294739821268975716162663306052858909500289219592060310551297322480112649057056185531676266039663602421928889.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607)

gp: K = bnfinit(y^45 - 279*y^43 + 35154*y^41 - 2656173*y^39 + 134805825*y^37 - 97712*y^36 - 4879171809*y^35 + 15188760*y^34 + 130534729785*y^33 - 1029724272*y^32 - 2640630159891*y^31 + 40401785400*y^30 + 40983205856724*y^29 - 1025408245788*y^28 - 492359066860125*y^27 + 17803202551128*y^26 + 4599619564395024*y^25 - 217725540576444*y^24 - 33445772871093027*y^23 + 1898451316158687*y^22 + 188849162457451617*y^21 - 11767696275794211*y^20 - 823332878002131822*y^19 + 50691889157596665*y^18 + 2745818611694276823*y^17 - 142361283528806580*y^16 - 6909518242574323155*y^15 + 208719160865886294*y^14 + 12865209764508264645*y^13 + 83974312294091439*y^12 - 17233171846493521500*y^11 - 1064212231177977606*y^10 + 15917394331044266446*y^9 + 2213346589608828033*y^8 - 9446941004466404511*y^7 - 2292874481029586463*y^6 + 3121786160210636919*y^5 + 1201035215446175088*y^4 - 355273519765875753*y^3 - 247333537481787396*y^2 - 40334300724080106*y - 1806463523301607, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607)

$ x^{45} - 279 x^{43} + 35154 x^{41} - 2656173 x^{39} + 134805825 x^{37} - 97712 x^{36} + \cdots - 18\!\cdots\!07 $ x^45 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607

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:	$131\!\cdots\!889$ 13198639294739821268975716162663306052858909500289219592060310551297322480112649057056185531676266039663602421928889 $\medspace = 3^{110}\cdot 31^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$361.60$	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:	$3^{22/9}31^{14/15}\approx 361.6046289801684$
Ramified primes:	$3$, $31$ 3, 31	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:	$837=3^{3}\cdot 31$
Dirichlet character group:	$\lbrace$$\chi_{837}(1,·)$, $\chi_{837}(388,·)$, $\chi_{837}(133,·)$, $\chi_{837}(769,·)$, $\chi_{837}(268,·)$, $\chi_{837}(142,·)$, $\chi_{837}(400,·)$, $\chi_{837}(280,·)$, $\chi_{837}(667,·)$, $\chi_{837}(412,·)$, $\chi_{837}(670,·)$, $\chi_{837}(160,·)$, $\chi_{837}(547,·)$, $\chi_{837}(421,·)$, $\chi_{837}(679,·)$, $\chi_{837}(391,·)$, $\chi_{837}(559,·)$, $\chi_{837}(691,·)$, $\chi_{837}(439,·)$, $\chi_{837}(826,·)$, $\chi_{837}(700,·)$, $\chi_{837}(190,·)$, $\chi_{837}(64,·)$, $\chi_{837}(193,·)$, $\chi_{837}(196,·)$, $\chi_{837}(76,·)$, $\chi_{837}(718,·)$, $\chi_{837}(721,·)$, $\chi_{837}(163,·)$, $\chi_{837}(469,·)$, $\chi_{837}(343,·)$, $\chi_{837}(472,·)$, $\chi_{837}(475,·)$, $\chi_{837}(442,·)$, $\chi_{837}(355,·)$, $\chi_{837}(490,·)$, $\chi_{837}(748,·)$, $\chi_{837}(109,·)$, $\chi_{837}(622,·)$, $\chi_{837}(751,·)$, $\chi_{837}(112,·)$, $\chi_{837}(754,·)$, $\chi_{837}(211,·)$, $\chi_{837}(121,·)$, $\chi_{837}(634,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31}a^{15}$, $\frac{1}{31}a^{16}$, $\frac{1}{31}a^{17}$, $\frac{1}{31}a^{18}$, $\frac{1}{31}a^{19}$, $\frac{1}{31}a^{20}$, $\frac{1}{31}a^{21}$, $\frac{1}{31}a^{22}$, $\frac{1}{31}a^{23}$, $\frac{1}{31}a^{24}$, $\frac{1}{31}a^{25}$, $\frac{1}{31}a^{26}$, $\frac{1}{1147}a^{27}+\frac{4}{1147}a^{26}+\frac{7}{1147}a^{25}-\frac{9}{1147}a^{24}+\frac{18}{1147}a^{23}+\frac{10}{1147}a^{22}+\frac{6}{1147}a^{21}+\frac{13}{1147}a^{20}-\frac{6}{1147}a^{19}+\frac{5}{1147}a^{18}+\frac{5}{1147}a^{17}+\frac{5}{1147}a^{16}+\frac{4}{1147}a^{15}-\frac{3}{37}a^{14}+\frac{14}{37}a^{13}+\frac{11}{37}a^{12}-\frac{13}{37}a^{11}-\frac{17}{37}a^{10}-\frac{16}{37}a^{9}+\frac{5}{37}a^{8}+\frac{1}{37}a^{7}+\frac{16}{37}a^{6}+\frac{12}{37}a^{5}-\frac{2}{37}a^{4}+\frac{6}{37}a^{3}+\frac{3}{37}a^{2}+\frac{4}{37}a$, $\frac{1}{1147}a^{28}-\frac{9}{1147}a^{26}+\frac{17}{1147}a^{24}+\frac{12}{1147}a^{23}+\frac{3}{1147}a^{22}-\frac{11}{1147}a^{21}+\frac{16}{1147}a^{20}-\frac{8}{1147}a^{19}-\frac{15}{1147}a^{18}-\frac{15}{1147}a^{17}-\frac{16}{1147}a^{16}+\frac{2}{1147}a^{15}-\frac{11}{37}a^{14}-\frac{8}{37}a^{13}+\frac{17}{37}a^{12}-\frac{2}{37}a^{11}+\frac{15}{37}a^{10}-\frac{5}{37}a^{9}+\frac{18}{37}a^{8}+\frac{12}{37}a^{7}-\frac{15}{37}a^{6}-\frac{13}{37}a^{5}+\frac{14}{37}a^{4}+\frac{16}{37}a^{3}-\frac{8}{37}a^{2}-\frac{16}{37}a$, $\frac{1}{1147}a^{29}-\frac{1}{1147}a^{26}+\frac{6}{1147}a^{25}+\frac{5}{1147}a^{24}+\frac{17}{1147}a^{23}+\frac{5}{1147}a^{22}-\frac{4}{1147}a^{21}-\frac{2}{1147}a^{20}+\frac{5}{1147}a^{19}-\frac{7}{1147}a^{18}-\frac{8}{1147}a^{17}+\frac{10}{1147}a^{16}-\frac{9}{1147}a^{15}+\frac{2}{37}a^{14}-\frac{5}{37}a^{13}-\frac{14}{37}a^{12}+\frac{9}{37}a^{11}-\frac{10}{37}a^{10}-\frac{15}{37}a^{9}-\frac{17}{37}a^{8}-\frac{6}{37}a^{7}-\frac{17}{37}a^{6}+\frac{11}{37}a^{5}-\frac{2}{37}a^{4}+\frac{9}{37}a^{3}+\frac{11}{37}a^{2}-\frac{1}{37}a$, $\frac{1}{35557}a^{30}-\frac{14}{1147}a^{26}-\frac{2}{1147}a^{25}+\frac{11}{1147}a^{24}-\frac{10}{1147}a^{23}-\frac{1}{1147}a^{22}-\frac{13}{1147}a^{21}-\frac{3}{1147}a^{20}-\frac{4}{1147}a^{19}-\frac{18}{1147}a^{18}+\frac{16}{1147}a^{17}+\frac{13}{1147}a^{16}-\frac{11}{1147}a^{15}-\frac{11}{37}a^{14}+\frac{9}{37}a^{12}+\frac{10}{37}a^{11}-\frac{7}{37}a^{10}-\frac{13}{37}a^{9}-\frac{6}{37}a^{8}+\frac{15}{37}a^{7}+\frac{14}{37}a^{6}-\frac{14}{37}a^{5}+\frac{5}{37}a^{4}-\frac{9}{37}a^{3}+\frac{12}{37}a^{2}-\frac{13}{37}a$, $\frac{1}{35557}a^{31}+\frac{17}{1147}a^{26}-\frac{2}{1147}a^{25}+\frac{12}{1147}a^{24}-\frac{8}{1147}a^{23}+\frac{16}{1147}a^{22}+\frac{7}{1147}a^{21}-\frac{7}{1147}a^{20}+\frac{9}{1147}a^{19}+\frac{12}{1147}a^{18}+\frac{9}{1147}a^{17}-\frac{15}{1147}a^{16}+\frac{11}{1147}a^{15}-\frac{5}{37}a^{14}-\frac{17}{37}a^{13}+\frac{16}{37}a^{12}-\frac{4}{37}a^{11}+\frac{8}{37}a^{10}-\frac{8}{37}a^{9}+\frac{11}{37}a^{8}-\frac{9}{37}a^{7}-\frac{12}{37}a^{6}-\frac{12}{37}a^{5}-\frac{15}{37}a^{3}-\frac{8}{37}a^{2}-\frac{18}{37}a$, $\frac{1}{35557}a^{32}+\frac{4}{1147}a^{26}+\frac{4}{1147}a^{25}-\frac{3}{1147}a^{24}+\frac{6}{1147}a^{23}-\frac{15}{1147}a^{22}+\frac{2}{1147}a^{21}+\frac{10}{1147}a^{20}+\frac{3}{1147}a^{19}-\frac{2}{1147}a^{18}+\frac{11}{1147}a^{17}-\frac{1}{1147}a^{15}-\frac{3}{37}a^{14}-\frac{6}{37}a^{12}+\frac{7}{37}a^{11}-\frac{15}{37}a^{10}-\frac{13}{37}a^{9}+\frac{17}{37}a^{8}+\frac{8}{37}a^{7}+\frac{12}{37}a^{6}+\frac{18}{37}a^{5}-\frac{18}{37}a^{4}+\frac{1}{37}a^{3}+\frac{5}{37}a^{2}+\frac{6}{37}a$, $\frac{1}{35557}a^{33}-\frac{12}{1147}a^{26}+\frac{6}{1147}a^{25}+\frac{5}{1147}a^{24}-\frac{13}{1147}a^{23}-\frac{1}{1147}a^{22}-\frac{14}{1147}a^{21}-\frac{12}{1147}a^{20}-\frac{15}{1147}a^{19}-\frac{9}{1147}a^{18}+\frac{17}{1147}a^{17}+\frac{16}{1147}a^{16}+\frac{2}{1147}a^{15}+\frac{12}{37}a^{14}+\frac{12}{37}a^{13}+\frac{18}{37}a^{10}+\frac{7}{37}a^{9}-\frac{12}{37}a^{8}+\frac{8}{37}a^{7}-\frac{9}{37}a^{6}+\frac{8}{37}a^{5}+\frac{9}{37}a^{4}+\frac{18}{37}a^{3}-\frac{6}{37}a^{2}-\frac{16}{37}a$, $\frac{1}{35557}a^{34}+\frac{17}{1147}a^{26}+\frac{15}{1147}a^{25}-\frac{10}{1147}a^{24}-\frac{7}{1147}a^{23}-\frac{5}{1147}a^{22}-\frac{14}{1147}a^{21}-\frac{7}{1147}a^{20}-\frac{7}{1147}a^{19}+\frac{3}{1147}a^{18}+\frac{2}{1147}a^{17}-\frac{12}{1147}a^{16}+\frac{13}{1147}a^{15}+\frac{13}{37}a^{14}-\frac{17}{37}a^{13}-\frac{16}{37}a^{12}+\frac{10}{37}a^{11}-\frac{12}{37}a^{10}+\frac{18}{37}a^{9}-\frac{6}{37}a^{8}+\frac{3}{37}a^{7}+\frac{15}{37}a^{6}+\frac{5}{37}a^{5}-\frac{6}{37}a^{4}-\frac{8}{37}a^{3}-\frac{17}{37}a^{2}+\frac{11}{37}a$, $\frac{1}{35557}a^{35}-\frac{16}{1147}a^{26}-\frac{18}{1147}a^{25}-\frac{2}{1147}a^{24}-\frac{15}{1147}a^{23}+\frac{1}{1147}a^{22}+\frac{2}{1147}a^{21}-\frac{6}{1147}a^{20}-\frac{6}{1147}a^{19}-\frac{9}{1147}a^{18}+\frac{14}{1147}a^{17}+\frac{2}{1147}a^{16}+\frac{2}{1147}a^{15}-\frac{3}{37}a^{14}+\frac{5}{37}a^{13}+\frac{8}{37}a^{12}-\frac{13}{37}a^{11}+\frac{11}{37}a^{10}+\frac{7}{37}a^{9}-\frac{8}{37}a^{8}-\frac{2}{37}a^{7}-\frac{8}{37}a^{6}+\frac{12}{37}a^{5}-\frac{11}{37}a^{4}-\frac{8}{37}a^{3}-\frac{3}{37}a^{2}+\frac{6}{37}a$, $\frac{1}{177785}a^{36}-\frac{1}{177785}a^{34}-\frac{2}{177785}a^{32}-\frac{1}{177785}a^{30}-\frac{1}{5735}a^{28}+\frac{2}{5735}a^{27}+\frac{3}{1147}a^{26}-\frac{8}{5735}a^{25}-\frac{4}{5735}a^{24}-\frac{3}{1147}a^{23}-\frac{44}{5735}a^{22}+\frac{5}{1147}a^{21}-\frac{57}{5735}a^{20}-\frac{6}{1147}a^{19}+\frac{64}{5735}a^{18}+\frac{6}{1147}a^{17}-\frac{4}{5735}a^{16}-\frac{12}{1147}a^{15}+\frac{8}{37}a^{14}+\frac{26}{185}a^{13}+\frac{2}{185}a^{12}+\frac{4}{185}a^{11}+\frac{68}{185}a^{10}-\frac{17}{37}a^{9}-\frac{63}{185}a^{8}-\frac{73}{185}a^{7}+\frac{77}{185}a^{6}+\frac{6}{185}a^{5}+\frac{16}{185}a^{4}-\frac{7}{185}a^{3}+\frac{26}{185}a^{2}-\frac{33}{185}a-\frac{1}{5}$, $\frac{1}{177785}a^{37}-\frac{1}{177785}a^{35}-\frac{2}{177785}a^{33}-\frac{1}{177785}a^{31}-\frac{1}{5735}a^{29}+\frac{2}{5735}a^{28}-\frac{68}{5735}a^{26}+\frac{76}{5735}a^{25}-\frac{13}{1147}a^{24}+\frac{56}{5735}a^{23}+\frac{12}{1147}a^{22}+\frac{38}{5735}a^{21}-\frac{8}{1147}a^{20}-\frac{1}{185}a^{19}-\frac{9}{1147}a^{18}-\frac{79}{5735}a^{17}+\frac{10}{1147}a^{16}+\frac{14}{1147}a^{15}+\frac{71}{185}a^{14}-\frac{23}{185}a^{13}+\frac{24}{185}a^{12}+\frac{78}{185}a^{11}-\frac{3}{37}a^{10}-\frac{8}{185}a^{9}+\frac{1}{5}a^{8}+\frac{62}{185}a^{7}-\frac{49}{185}a^{6}+\frac{21}{185}a^{5}+\frac{23}{185}a^{4}-\frac{64}{185}a^{3}-\frac{78}{185}a^{2}+\frac{88}{185}a$, $\frac{1}{177785}a^{38}+\frac{2}{177785}a^{34}+\frac{2}{177785}a^{32}-\frac{2}{177785}a^{30}+\frac{2}{5735}a^{29}-\frac{1}{5735}a^{28}-\frac{1}{5735}a^{27}+\frac{36}{5735}a^{26}+\frac{47}{5735}a^{25}-\frac{83}{5735}a^{24}-\frac{3}{1147}a^{23}-\frac{41}{5735}a^{22}-\frac{15}{1147}a^{21}-\frac{58}{5735}a^{20}-\frac{10}{1147}a^{19}-\frac{8}{1147}a^{18}+\frac{5}{1147}a^{17}-\frac{19}{5735}a^{16}-\frac{89}{5735}a^{15}-\frac{88}{185}a^{14}-\frac{10}{37}a^{13}+\frac{6}{37}a^{12}+\frac{84}{185}a^{11}+\frac{18}{37}a^{10}+\frac{27}{185}a^{9}+\frac{14}{185}a^{8}+\frac{78}{185}a^{7}+\frac{28}{185}a^{6}-\frac{51}{185}a^{5}+\frac{1}{5}a^{4}+\frac{54}{185}a^{2}-\frac{78}{185}a-\frac{1}{5}$, $\frac{1}{177785}a^{39}+\frac{2}{177785}a^{35}+\frac{2}{177785}a^{33}-\frac{2}{177785}a^{31}+\frac{2}{177785}a^{30}-\frac{1}{5735}a^{29}-\frac{1}{5735}a^{28}+\frac{1}{5735}a^{27}+\frac{7}{5735}a^{26}-\frac{23}{5735}a^{25}+\frac{2}{1147}a^{24}-\frac{71}{5735}a^{23}+\frac{1}{1147}a^{22}-\frac{43}{5735}a^{21}+\frac{9}{1147}a^{20}+\frac{8}{1147}a^{19}+\frac{1}{1147}a^{18}-\frac{44}{5735}a^{17}+\frac{66}{5735}a^{16}+\frac{12}{5735}a^{15}-\frac{5}{37}a^{14}-\frac{18}{37}a^{13}+\frac{84}{185}a^{12}-\frac{11}{37}a^{11}-\frac{68}{185}a^{10}+\frac{59}{185}a^{9}+\frac{78}{185}a^{8}+\frac{18}{185}a^{7}+\frac{29}{185}a^{6}+\frac{87}{185}a^{5}-\frac{9}{37}a^{4}+\frac{14}{185}a^{3}+\frac{22}{185}a^{2}+\frac{48}{185}a$, $\frac{1}{6578045}a^{40}-\frac{13}{6578045}a^{39}+\frac{7}{6578045}a^{38}+\frac{16}{6578045}a^{37}-\frac{9}{6578045}a^{36}+\frac{63}{6578045}a^{35}-\frac{83}{6578045}a^{34}+\frac{17}{6578045}a^{33}+\frac{84}{6578045}a^{32}-\frac{53}{6578045}a^{31}+\frac{7}{1315609}a^{30}-\frac{10}{42439}a^{29}-\frac{3}{42439}a^{28}-\frac{15}{42439}a^{27}-\frac{316}{42439}a^{26}-\frac{83}{6845}a^{25}+\frac{51}{5735}a^{24}-\frac{2916}{212195}a^{23}-\frac{1521}{212195}a^{22}+\frac{2827}{212195}a^{21}-\frac{3234}{212195}a^{20}+\frac{1329}{212195}a^{19}+\frac{2242}{212195}a^{18}-\frac{1806}{212195}a^{17}-\frac{582}{42439}a^{16}+\frac{6}{6845}a^{15}+\frac{270}{1369}a^{14}+\frac{131}{1369}a^{13}+\frac{574}{1369}a^{12}+\frac{349}{6845}a^{11}-\frac{295}{1369}a^{10}-\frac{1023}{6845}a^{9}-\frac{2218}{6845}a^{8}+\frac{1741}{6845}a^{7}-\frac{17}{1369}a^{6}-\frac{3413}{6845}a^{5}+\frac{167}{1369}a^{4}-\frac{2292}{6845}a^{3}-\frac{119}{6845}a^{2}-\frac{2409}{6845}a-\frac{41}{185}$, $\frac{1}{13\!\cdots\!85}a^{41}-\frac{1230775312}{13\!\cdots\!85}a^{40}-\frac{34816014263}{27\!\cdots\!37}a^{39}-\frac{107580137303}{13\!\cdots\!85}a^{38}-\frac{242409401819}{13\!\cdots\!85}a^{37}+\frac{198361452252}{13\!\cdots\!85}a^{36}-\frac{1320794785182}{13\!\cdots\!85}a^{35}-\frac{1671691896446}{13\!\cdots\!85}a^{34}-\frac{71677086678}{27\!\cdots\!37}a^{33}-\frac{161919906387}{13\!\cdots\!85}a^{32}-\frac{1385642448179}{13\!\cdots\!85}a^{31}+\frac{561297954321}{13\!\cdots\!85}a^{30}+\frac{118422329023}{44\!\cdots\!35}a^{29}-\frac{286770882896}{896753902570727}a^{28}-\frac{1595507756192}{44\!\cdots\!35}a^{27}-\frac{19692159009509}{44\!\cdots\!35}a^{26}-\frac{107556220536}{24772207253335}a^{25}-\frac{45565044474123}{44\!\cdots\!35}a^{24}+\frac{416778824103}{121182959806855}a^{23}+\frac{11017637575436}{896753902570727}a^{22}+\frac{4292966060992}{44\!\cdots\!35}a^{21}-\frac{52500823995628}{44\!\cdots\!35}a^{20}-\frac{40533366707448}{44\!\cdots\!35}a^{19}+\frac{24058607055338}{44\!\cdots\!35}a^{18}+\frac{37091344076949}{44\!\cdots\!35}a^{17}-\frac{26509138846941}{44\!\cdots\!35}a^{16}-\frac{52840000318}{24772207253335}a^{15}+\frac{27619287363622}{144637726221085}a^{14}-\frac{42615046641079}{144637726221085}a^{13}-\frac{9419928297961}{28927545244217}a^{12}-\frac{4721133088911}{144637726221085}a^{11}+\frac{163703097014}{3909127735705}a^{10}-\frac{44135398049871}{144637726221085}a^{9}+\frac{15707239040461}{144637726221085}a^{8}+\frac{6324018564621}{28927545244217}a^{7}-\frac{421136956717}{144637726221085}a^{6}+\frac{151420495306}{3909127735705}a^{5}-\frac{66560625860169}{144637726221085}a^{4}-\frac{27501089560479}{144637726221085}a^{3}+\frac{37396192642881}{144637726221085}a^{2}-\frac{46292912861957}{144637726221085}a+\frac{1538268976477}{3909127735705}$, $\frac{1}{13\!\cdots\!85}a^{42}+\frac{775431069}{13\!\cdots\!85}a^{40}-\frac{1799042619}{37\!\cdots\!05}a^{39}+\frac{186226679139}{13\!\cdots\!85}a^{38}+\frac{100583084119}{13\!\cdots\!85}a^{37}-\frac{13865267063}{13\!\cdots\!85}a^{36}-\frac{59428856166}{27\!\cdots\!37}a^{35}+\frac{560493928413}{13\!\cdots\!85}a^{34}+\frac{1346581592268}{13\!\cdots\!85}a^{33}+\frac{950426196156}{13\!\cdots\!85}a^{32}-\frac{5035006678}{37\!\cdots\!05}a^{31}-\frac{85760287295}{27\!\cdots\!37}a^{30}+\frac{338880168346}{44\!\cdots\!35}a^{29}-\frac{53217223607}{144637726221085}a^{28}-\frac{19553314708}{144637726221085}a^{27}+\frac{56745774396256}{44\!\cdots\!35}a^{26}+\frac{54648846423006}{44\!\cdots\!35}a^{25}+\frac{69810833628004}{44\!\cdots\!35}a^{24}+\frac{10307876934532}{44\!\cdots\!35}a^{23}+\frac{5200319336343}{44\!\cdots\!35}a^{22}-\frac{39027670423979}{44\!\cdots\!35}a^{21}-\frac{71633836278144}{44\!\cdots\!35}a^{20}-\frac{29207802844693}{44\!\cdots\!35}a^{19}+\frac{36984477566129}{44\!\cdots\!35}a^{18}+\frac{9526626123784}{896753902570727}a^{17}-\frac{55993633251091}{44\!\cdots\!35}a^{16}+\frac{1934972111836}{896753902570727}a^{15}+\frac{48923645624823}{144637726221085}a^{14}-\frac{63555187589027}{144637726221085}a^{13}+\frac{15880538264462}{144637726221085}a^{12}-\frac{54711348422666}{144637726221085}a^{11}+\frac{4600433465674}{144637726221085}a^{10}+\frac{50219197762831}{144637726221085}a^{9}-\frac{362356276589}{28927545244217}a^{8}-\frac{10952372935154}{28927545244217}a^{7}-\frac{40828637437641}{144637726221085}a^{6}-\frac{5389235830118}{28927545244217}a^{5}-\frac{28122795124038}{144637726221085}a^{4}+\frac{21395982628111}{144637726221085}a^{3}+\frac{32974266613849}{144637726221085}a^{2}+\frac{26738835866371}{144637726221085}a+\frac{1137205319916}{3909127735705}$, $\frac{1}{99\!\cdots\!15}a^{43}+\frac{45}{19\!\cdots\!03}a^{42}-\frac{28}{19\!\cdots\!03}a^{41}+\frac{4271314659917}{99\!\cdots\!15}a^{40}-\frac{88766725515574}{99\!\cdots\!15}a^{39}+\frac{1985258839343}{32\!\cdots\!65}a^{38}+\frac{256690586318294}{99\!\cdots\!15}a^{37}-\frac{30811871437707}{19\!\cdots\!03}a^{36}+\frac{427351301240769}{99\!\cdots\!15}a^{35}-\frac{12\!\cdots\!24}{99\!\cdots\!15}a^{34}+\frac{890135440327383}{99\!\cdots\!15}a^{33}+\frac{500182849683144}{99\!\cdots\!15}a^{32}-\frac{208133556765032}{19\!\cdots\!03}a^{31}-\frac{332763834004964}{99\!\cdots\!15}a^{30}+\frac{237658999435711}{64\!\cdots\!13}a^{29}+\frac{12\!\cdots\!48}{32\!\cdots\!65}a^{28}+\frac{12549357912668}{87\!\cdots\!45}a^{27}-\frac{31\!\cdots\!21}{32\!\cdots\!65}a^{26}-\frac{97\!\cdots\!73}{64\!\cdots\!13}a^{25}+\frac{41\!\cdots\!12}{64\!\cdots\!13}a^{24}+\frac{26\!\cdots\!88}{32\!\cdots\!65}a^{23}-\frac{22\!\cdots\!63}{32\!\cdots\!65}a^{22}-\frac{56\!\cdots\!81}{64\!\cdots\!13}a^{21}+\frac{17\!\cdots\!19}{64\!\cdots\!13}a^{20}-\frac{86\!\cdots\!02}{64\!\cdots\!13}a^{19}-\frac{36\!\cdots\!89}{32\!\cdots\!65}a^{18}+\frac{23\!\cdots\!53}{32\!\cdots\!65}a^{17}+\frac{16\!\cdots\!01}{32\!\cdots\!65}a^{16}+\frac{33\!\cdots\!36}{32\!\cdots\!65}a^{15}+\frac{671053025440255}{20\!\cdots\!23}a^{14}+\frac{43\!\cdots\!67}{10\!\cdots\!15}a^{13}-\frac{44\!\cdots\!21}{10\!\cdots\!15}a^{12}+\frac{30\!\cdots\!79}{10\!\cdots\!15}a^{11}-\frac{51\!\cdots\!87}{10\!\cdots\!15}a^{10}+\frac{35\!\cdots\!79}{10\!\cdots\!15}a^{9}+\frac{50\!\cdots\!61}{10\!\cdots\!15}a^{8}-\frac{88\!\cdots\!31}{10\!\cdots\!15}a^{7}+\frac{36\!\cdots\!12}{20\!\cdots\!23}a^{6}-\frac{13\!\cdots\!59}{10\!\cdots\!15}a^{5}+\frac{44\!\cdots\!21}{10\!\cdots\!15}a^{4}+\frac{12\!\cdots\!60}{20\!\cdots\!23}a^{3}+\frac{46\!\cdots\!21}{10\!\cdots\!15}a^{2}-\frac{982456255863069}{10\!\cdots\!15}a-\frac{10\!\cdots\!99}{28\!\cdots\!95}$, $\frac{1}{20\!\cdots\!55}a^{44}+\frac{29\!\cdots\!35}{41\!\cdots\!71}a^{43}-\frac{38\!\cdots\!27}{20\!\cdots\!55}a^{42}-\frac{51\!\cdots\!37}{20\!\cdots\!55}a^{41}-\frac{31\!\cdots\!42}{67\!\cdots\!05}a^{40}-\frac{41\!\cdots\!54}{41\!\cdots\!71}a^{39}-\frac{95\!\cdots\!30}{41\!\cdots\!71}a^{38}-\frac{39\!\cdots\!01}{20\!\cdots\!55}a^{37}-\frac{31\!\cdots\!93}{20\!\cdots\!55}a^{36}-\frac{30\!\cdots\!70}{41\!\cdots\!71}a^{35}+\frac{11\!\cdots\!78}{20\!\cdots\!55}a^{34}+\frac{26\!\cdots\!48}{20\!\cdots\!55}a^{33}+\frac{86\!\cdots\!99}{20\!\cdots\!55}a^{32}+\frac{30\!\cdots\!85}{41\!\cdots\!71}a^{31}-\frac{31\!\cdots\!94}{20\!\cdots\!55}a^{30}-\frac{67\!\cdots\!01}{67\!\cdots\!05}a^{29}+\frac{46\!\cdots\!30}{13\!\cdots\!41}a^{28}+\frac{23\!\cdots\!03}{67\!\cdots\!05}a^{27}-\frac{88\!\cdots\!21}{67\!\cdots\!05}a^{26}+\frac{10\!\cdots\!46}{67\!\cdots\!05}a^{25}-\frac{32\!\cdots\!99}{13\!\cdots\!41}a^{24}+\frac{91\!\cdots\!14}{67\!\cdots\!05}a^{23}-\frac{78\!\cdots\!23}{67\!\cdots\!05}a^{22}-\frac{36\!\cdots\!97}{13\!\cdots\!41}a^{21}+\frac{25\!\cdots\!47}{67\!\cdots\!05}a^{20}+\frac{85\!\cdots\!43}{67\!\cdots\!05}a^{19}+\frac{63\!\cdots\!59}{67\!\cdots\!05}a^{18}-\frac{17\!\cdots\!56}{67\!\cdots\!05}a^{17}-\frac{49\!\cdots\!59}{13\!\cdots\!41}a^{16}+\frac{33\!\cdots\!96}{67\!\cdots\!05}a^{15}+\frac{53\!\cdots\!48}{21\!\cdots\!55}a^{14}-\frac{14\!\cdots\!05}{43\!\cdots\!11}a^{13}-\frac{34\!\cdots\!72}{43\!\cdots\!11}a^{12}+\frac{92\!\cdots\!53}{21\!\cdots\!55}a^{11}-\frac{84\!\cdots\!98}{21\!\cdots\!55}a^{10}+\frac{79\!\cdots\!99}{21\!\cdots\!55}a^{9}+\frac{42\!\cdots\!96}{21\!\cdots\!55}a^{8}-\frac{23\!\cdots\!09}{21\!\cdots\!55}a^{7}-\frac{98\!\cdots\!97}{21\!\cdots\!55}a^{6}+\frac{16\!\cdots\!39}{43\!\cdots\!11}a^{5}-\frac{89\!\cdots\!97}{21\!\cdots\!55}a^{4}+\frac{10\!\cdots\!97}{21\!\cdots\!55}a^{3}+\frac{46\!\cdots\!58}{21\!\cdots\!55}a^{2}-\frac{37\!\cdots\!50}{43\!\cdots\!11}a+\frac{98\!\cdots\!12}{35\!\cdots\!15}$ 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, 1/31*a^15, 1/31*a^16, 1/31*a^17, 1/31*a^18, 1/31*a^19, 1/31*a^20, 1/31*a^21, 1/31*a^22, 1/31*a^23, 1/31*a^24, 1/31*a^25, 1/31*a^26, 1/1147*a^27 + 4/1147*a^26 + 7/1147*a^25 - 9/1147*a^24 + 18/1147*a^23 + 10/1147*a^22 + 6/1147*a^21 + 13/1147*a^20 - 6/1147*a^19 + 5/1147*a^18 + 5/1147*a^17 + 5/1147*a^16 + 4/1147*a^15 - 3/37*a^14 + 14/37*a^13 + 11/37*a^12 - 13/37*a^11 - 17/37*a^10 - 16/37*a^9 + 5/37*a^8 + 1/37*a^7 + 16/37*a^6 + 12/37*a^5 - 2/37*a^4 + 6/37*a^3 + 3/37*a^2 + 4/37*a, 1/1147*a^28 - 9/1147*a^26 + 17/1147*a^24 + 12/1147*a^23 + 3/1147*a^22 - 11/1147*a^21 + 16/1147*a^20 - 8/1147*a^19 - 15/1147*a^18 - 15/1147*a^17 - 16/1147*a^16 + 2/1147*a^15 - 11/37*a^14 - 8/37*a^13 + 17/37*a^12 - 2/37*a^11 + 15/37*a^10 - 5/37*a^9 + 18/37*a^8 + 12/37*a^7 - 15/37*a^6 - 13/37*a^5 + 14/37*a^4 + 16/37*a^3 - 8/37*a^2 - 16/37*a, 1/1147*a^29 - 1/1147*a^26 + 6/1147*a^25 + 5/1147*a^24 + 17/1147*a^23 + 5/1147*a^22 - 4/1147*a^21 - 2/1147*a^20 + 5/1147*a^19 - 7/1147*a^18 - 8/1147*a^17 + 10/1147*a^16 - 9/1147*a^15 + 2/37*a^14 - 5/37*a^13 - 14/37*a^12 + 9/37*a^11 - 10/37*a^10 - 15/37*a^9 - 17/37*a^8 - 6/37*a^7 - 17/37*a^6 + 11/37*a^5 - 2/37*a^4 + 9/37*a^3 + 11/37*a^2 - 1/37*a, 1/35557*a^30 - 14/1147*a^26 - 2/1147*a^25 + 11/1147*a^24 - 10/1147*a^23 - 1/1147*a^22 - 13/1147*a^21 - 3/1147*a^20 - 4/1147*a^19 - 18/1147*a^18 + 16/1147*a^17 + 13/1147*a^16 - 11/1147*a^15 - 11/37*a^14 + 9/37*a^12 + 10/37*a^11 - 7/37*a^10 - 13/37*a^9 - 6/37*a^8 + 15/37*a^7 + 14/37*a^6 - 14/37*a^5 + 5/37*a^4 - 9/37*a^3 + 12/37*a^2 - 13/37*a, 1/35557*a^31 + 17/1147*a^26 - 2/1147*a^25 + 12/1147*a^24 - 8/1147*a^23 + 16/1147*a^22 + 7/1147*a^21 - 7/1147*a^20 + 9/1147*a^19 + 12/1147*a^18 + 9/1147*a^17 - 15/1147*a^16 + 11/1147*a^15 - 5/37*a^14 - 17/37*a^13 + 16/37*a^12 - 4/37*a^11 + 8/37*a^10 - 8/37*a^9 + 11/37*a^8 - 9/37*a^7 - 12/37*a^6 - 12/37*a^5 - 15/37*a^3 - 8/37*a^2 - 18/37*a, 1/35557*a^32 + 4/1147*a^26 + 4/1147*a^25 - 3/1147*a^24 + 6/1147*a^23 - 15/1147*a^22 + 2/1147*a^21 + 10/1147*a^20 + 3/1147*a^19 - 2/1147*a^18 + 11/1147*a^17 - 1/1147*a^15 - 3/37*a^14 - 6/37*a^12 + 7/37*a^11 - 15/37*a^10 - 13/37*a^9 + 17/37*a^8 + 8/37*a^7 + 12/37*a^6 + 18/37*a^5 - 18/37*a^4 + 1/37*a^3 + 5/37*a^2 + 6/37*a, 1/35557*a^33 - 12/1147*a^26 + 6/1147*a^25 + 5/1147*a^24 - 13/1147*a^23 - 1/1147*a^22 - 14/1147*a^21 - 12/1147*a^20 - 15/1147*a^19 - 9/1147*a^18 + 17/1147*a^17 + 16/1147*a^16 + 2/1147*a^15 + 12/37*a^14 + 12/37*a^13 + 18/37*a^10 + 7/37*a^9 - 12/37*a^8 + 8/37*a^7 - 9/37*a^6 + 8/37*a^5 + 9/37*a^4 + 18/37*a^3 - 6/37*a^2 - 16/37*a, 1/35557*a^34 + 17/1147*a^26 + 15/1147*a^25 - 10/1147*a^24 - 7/1147*a^23 - 5/1147*a^22 - 14/1147*a^21 - 7/1147*a^20 - 7/1147*a^19 + 3/1147*a^18 + 2/1147*a^17 - 12/1147*a^16 + 13/1147*a^15 + 13/37*a^14 - 17/37*a^13 - 16/37*a^12 + 10/37*a^11 - 12/37*a^10 + 18/37*a^9 - 6/37*a^8 + 3/37*a^7 + 15/37*a^6 + 5/37*a^5 - 6/37*a^4 - 8/37*a^3 - 17/37*a^2 + 11/37*a, 1/35557*a^35 - 16/1147*a^26 - 18/1147*a^25 - 2/1147*a^24 - 15/1147*a^23 + 1/1147*a^22 + 2/1147*a^21 - 6/1147*a^20 - 6/1147*a^19 - 9/1147*a^18 + 14/1147*a^17 + 2/1147*a^16 + 2/1147*a^15 - 3/37*a^14 + 5/37*a^13 + 8/37*a^12 - 13/37*a^11 + 11/37*a^10 + 7/37*a^9 - 8/37*a^8 - 2/37*a^7 - 8/37*a^6 + 12/37*a^5 - 11/37*a^4 - 8/37*a^3 - 3/37*a^2 + 6/37*a, 1/177785*a^36 - 1/177785*a^34 - 2/177785*a^32 - 1/177785*a^30 - 1/5735*a^28 + 2/5735*a^27 + 3/1147*a^26 - 8/5735*a^25 - 4/5735*a^24 - 3/1147*a^23 - 44/5735*a^22 + 5/1147*a^21 - 57/5735*a^20 - 6/1147*a^19 + 64/5735*a^18 + 6/1147*a^17 - 4/5735*a^16 - 12/1147*a^15 + 8/37*a^14 + 26/185*a^13 + 2/185*a^12 + 4/185*a^11 + 68/185*a^10 - 17/37*a^9 - 63/185*a^8 - 73/185*a^7 + 77/185*a^6 + 6/185*a^5 + 16/185*a^4 - 7/185*a^3 + 26/185*a^2 - 33/185*a - 1/5, 1/177785*a^37 - 1/177785*a^35 - 2/177785*a^33 - 1/177785*a^31 - 1/5735*a^29 + 2/5735*a^28 - 68/5735*a^26 + 76/5735*a^25 - 13/1147*a^24 + 56/5735*a^23 + 12/1147*a^22 + 38/5735*a^21 - 8/1147*a^20 - 1/185*a^19 - 9/1147*a^18 - 79/5735*a^17 + 10/1147*a^16 + 14/1147*a^15 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91140889320112281289714394325530562868826211648664799295923243971340327342207409083029703677793668883657559457725967892397174460194331819571611006215456995354426446561852641395007613795361342750798405377430968387194346418517883588655702976590785693134728262814/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^23 - 78835709687781321659072933455076386296283096652013203344792627596415459045668952185331453207373074118874026789856365660465005636216977626393807267321109283304500275300662255442730095852237753978011147198879700776892520272647193275521242371621239279683844031723/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^22 - 3693004081173325456436993287933474639250744164649192727528381487483276543448623467595955476089811182150192507027821548352990023844302323499371540715613785604108938807376359771051576055872457886429334322455212764893164750532847720041874786957275874477048347397/1341185227184303028163804621525110883770189536638842070976150300399578456382088555295467130973196766672956988845654202463570358458733639733572499209738897097207499921966127484009721307631539799775789135718541348785691284913268896164225012159799721942539959013841*a^21 + 25021194421450602681099018884319032504456808767644841397614099329676583128896907753450312998096231147563757771483673691048046165155930971665054321825923892629950028293204178567771797484980770846100179166501922327844810260871457315750675705574413859643470992047/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^20 + 8517505140201209814729747836485851291242972708438164991460268143712602618064180957298531354398663443656788411765479810847235000062558157645956281394982953617029745975577565535953459329867123321425462192484229278331115737412846643461120985697175270991025365443/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^19 + 63044199470067124750418466845104372456027806756781680458385084915796008728229022017627880463061807138618301261164401004954678176340102190915851264742591222780731068194566528833642373648224853632029109146725723480984740198059226182783750510880802108328156734059/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^18 - 17928815832956981943805720154000769386609646736071237079127526892694406061207262702527156218614002342218359149772079562224300770129357418048550390571456932165910506438431664019586728511005489127011028310678903407087026140312310776157833402638392493301666403156/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^17 - 4924201133963329981887001018242011449971891367333364533058836697138583579093671018826475768086684181418178666131949148190117711846586836643066539033252522530062046397288863359419978143010353982542072926913749799456048370723883256273454582630956417724042735359/1341185227184303028163804621525110883770189536638842070976150300399578456382088555295467130973196766672956988845654202463570358458733639733572499209738897097207499921966127484009721307631539799775789135718541348785691284913268896164225012159799721942539959013841*a^16 + 33833116060146136850505897911326211298345050322856936110753951355933500047745310800966590933238178591624309773450617929759659504585781878431643021679407702675761185489463765330552318067625733165947899222613692502116428263250765138577330072772352098685213668396/6705926135921515140819023107625554418850947683194210354880751501997892281910442776477335654865983833364784944228271012317851792293668198667862496048694485486037499609830637420048606538157698998878945678592706743928456424566344480821125060798998609712699795069205*a^15 + 53916499495702478338109905480184833437890704201345154842331240500505783061266947851027033032456853706062936825142247237706680297288833476027439093635251934910502148609088354795965131191063820474243720541343145799403477412231320815011325691175596999197918047548/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^14 - 14253571458571037549956554564928731406021167145232814447958580626070922126739440963559335312446390191680109565085703934595672932164505183288834163049579132670814218715027696791771490011380911302129821854779400890355613057870910860834066114515793399585756498305/43264039586590420263348536178229383347425468923833615192779041948373498592970598557918294547522476344288935124053361369792592208346246443018467716443190228942177416837617015613216816375210961283089972119952946735022299513331254714975000392251603933630321258511*a^13 - 3438345004034845042260460689402702287386667523293438715954548494935421097766975582432422671210868522084517762738989775581000880083586850305253030606021692309788659249999531637783268396055217924320289157520907300561043280090448199822786017532640613089592299672/43264039586590420263348536178229383347425468923833615192779041948373498592970598557918294547522476344288935124053361369792592208346246443018467716443190228942177416837617015613216816375210961283089972119952946735022299513331254714975000392251603933630321258511*a^12 + 92302022077754107505291136136354921160165683102016736467624960043754490449241798956879768482367851836185411732125930448525145741243551491057389961593161022480425753680645760104172875150126621296160657340182372859035565155406069436511383355349575282208105938453/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^11 - 84268132784292920055092231768533467381534305610890339406466391089535981940995740496474854473729142185652393518662459807046101676291141721117396767957395261456731087099287356715463552985374451114411295816454999069746447034476842464171346108331203399932128713698/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^10 + 79429618751173329984986386658653525253533674532169493752839223625915680184101324289271250634962247361592247354172720353033666061955263809817394082739383958632923351053071319165407246877224202736868021111068050950894139877249440100886997958142209694776860450499/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^9 + 42438725849920601600495124178923623851009074385847060082824447202334035947806800326314397208317956924285979615364059253971269580852242857662008243418959427623247479342329688228529898396034577335680274661937982587712148871676328100173966741081996637200892144996/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^8 - 23212416686464288771307489112158143917395976321965895145530343003156364927162689163600913616556800166471878925030394832224906136265632801093047794850676278402374656161013289847823833472456912626404945320765832536832702675973402510698265670328604251802899661509/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^7 - 98525616195934882872292567031017728933389568085195517792604326486824095895151933175904093498985313701525221623575725967793357818264645622515452092428275908356113453371278444971418152801423352842725535735888567212205390967888733550130496228486890603017985942797/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^6 + 16797070855668332137576936461776291983977737514554744100425726151385324621699214648389540775896973954633583446528825664905421947430639266324131235676071575605691102358687513521258902702189951643347924594038673722154153741275884389243947280931837733797884288939/43264039586590420263348536178229383347425468923833615192779041948373498592970598557918294547522476344288935124053361369792592208346246443018467716443190228942177416837617015613216816375210961283089972119952946735022299513331254714975000392251603933630321258511*a^5 - 89766058587183908382696284013866937603145808556428515493392423250320462708788294465095535206265938768236938982651084055379913227503926716793220467194405546254515076753328685131071617064091704595139001667786784450413088221356764003728536447100804151384098906497/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^4 + 102941435588634318373584799969861931096200486295002776632077166737819255685846573950117603660924911721157062085516780371404235106369879006841323470344085625184623616677214292912141545805449128583933041940451505693778096335788534992197912352962575258199641986097/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^3 + 46261497042842817317360410802554502480848299110865283731585634328069473174402271413714576511934123453532022470217402332771759378539099418497608958531640600383872634524587913866918301125290351468531386773281254648600729772941696738981891931264103548733571179558/216320197932952101316742680891146916737127344619168075963895209741867492964852992789591472737612381721444675620266806848962961041731232215092338582215951144710887084188085078066084081876054806415449860599764733675111497566656273574875001961258019668151606292555*a^2 - 3740438459267414116953780874426198556892474713635277007760689484007308570551679601364006253117733410106645167056109145478499577345723797305001253728545479057290668369784687424512910383592965048944662155486823853130057348398914557157795837139095809414075968650/43264039586590420263348536178229383347425468923833615192779041948373498592970598557918294547522476344288935124053361369792592208346246443018467716443190228942177416837617015613216816375210961283089972119952946735022299513331254714975000392251603933630321258511*a + 985825550543590055631442084870094128072259625359648862624072296897995021711649916565653441383951755453691283146012160549625502018606542743037455153093052701331901920340597398997096949278300467783290848184982455903030579625605017185846061372256688412/3567409434784706910979220535929109944356064177534038534055078970603285345069871319579493598909385859753834171607900094776124437974091644891610689867204905201567121420318979654508398499509715365385232899009302188906555639717014933182778393956742751715

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$37$

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 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607)

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 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607, 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 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607);

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 - 279*x^43 + 35154*x^41 - 2656173*x^39 + 134805825*x^37 - 97712*x^36 - 4879171809*x^35 + 15188760*x^34 + 130534729785*x^33 - 1029724272*x^32 - 2640630159891*x^31 + 40401785400*x^30 + 40983205856724*x^29 - 1025408245788*x^28 - 492359066860125*x^27 + 17803202551128*x^26 + 4599619564395024*x^25 - 217725540576444*x^24 - 33445772871093027*x^23 + 1898451316158687*x^22 + 188849162457451617*x^21 - 11767696275794211*x^20 - 823332878002131822*x^19 + 50691889157596665*x^18 + 2745818611694276823*x^17 - 142361283528806580*x^16 - 6909518242574323155*x^15 + 208719160865886294*x^14 + 12865209764508264645*x^13 + 83974312294091439*x^12 - 17233171846493521500*x^11 - 1064212231177977606*x^10 + 15917394331044266446*x^9 + 2213346589608828033*x^8 - 9446941004466404511*x^7 - 2292874481029586463*x^6 + 3121786160210636919*x^5 + 1201035215446175088*x^4 - 355273519765875753*x^3 - 247333537481787396*x^2 - 40334300724080106*x - 1806463523301607);

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

$C_{45}$ (as 45T1):

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)

A cyclic group of order 45

The 45 conjugacy class representatives for $C_{45}$

Character table for $C_{45}$

Intermediate fields

$\Q(\zeta_{9})^+$, 5.5.923521.1, 9.9.27850805916667920729.1, 15.15.2746410307762150989067078161.1

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	$45$	R	${\href{/padicField/5.9.0.1}{9} }^{5}$	$45$	$45$	$45$	${\href{/padicField/17.5.0.1}{5} }^{9}$	$15^{3}$	$45$	$45$	R	${\href{/padicField/37.1.0.1}{1} }^{45}$	$45$	$45$	$45$	$15^{3}$	$45$

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 $45$	$9$	$5$	$110$
$31$ 31		Deg $45$	$15$	$3$	$42$

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