LMFDB - Number field 44.0.21142110273569853339470849219791293286477383243143882449296109338159240255146818984969.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625)

gp: K = bnfinit(y^44 - y^43 - 4*y^42 + 9*y^41 + 11*y^40 - 56*y^39 + y^38 + 279*y^37 - 284*y^36 - 1111*y^35 + 2531*y^34 + 3024*y^33 - 15679*y^32 + 559*y^31 + 77836*y^30 - 80631*y^29 - 308549*y^28 + 711704*y^27 + 831041*y^26 - 4389561*y^25 + 234356*y^24 + 21713449*y^23 - 22885229*y^22 + 108567245*y^21 + 5858900*y^20 - 548695125*y^19 + 519400625*y^18 + 2224075000*y^17 - 4821078125*y^16 - 6299296875*y^15 + 30404687500*y^14 + 1091796875*y^13 - 153115234375*y^12 + 147656250000*y^11 + 617919921875*y^10 - 1356201171875*y^9 - 1733398437500*y^8 + 8514404296875*y^7 + 152587890625*y^6 - 42724609375000*y^5 + 41961669921875*y^4 + 171661376953125*y^3 - 381469726562500*y^2 - 476837158203125*y + 2384185791015625, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625)

$ x^{44} - x^{43} - 4 x^{42} + 9 x^{41} + 11 x^{40} - 56 x^{39} + x^{38} + 279 x^{37} + \cdots + 23\!\cdots\!25 $ x^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$44$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 22]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$211\!\cdots\!969$ 21142110273569853339470849219791293286477383243143882449296109338159240255146818984969 $\medspace = 19^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$86.94$	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:	$19^{1/2}23^{21/22}\approx 86.93765400716107$
Ramified primes:	$19$, $23$ 19, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$44$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$437=19\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(132,·)$, $\chi_{437}(134,·)$, $\chi_{437}(265,·)$, $\chi_{437}(267,·)$, $\chi_{437}(398,·)$, $\chi_{437}(400,·)$, $\chi_{437}(18,·)$, $\chi_{437}(20,·)$, $\chi_{437}(151,·)$, $\chi_{437}(153,·)$, $\chi_{437}(284,·)$, $\chi_{437}(286,·)$, $\chi_{437}(417,·)$, $\chi_{437}(419,·)$, $\chi_{437}(37,·)$, $\chi_{437}(39,·)$, $\chi_{437}(170,·)$, $\chi_{437}(172,·)$, $\chi_{437}(303,·)$, $\chi_{437}(305,·)$, $\chi_{437}(436,·)$, $\chi_{437}(56,·)$, $\chi_{437}(58,·)$, $\chi_{437}(189,·)$, $\chi_{437}(191,·)$, $\chi_{437}(324,·)$, $\chi_{437}(75,·)$, $\chi_{437}(77,·)$, $\chi_{437}(208,·)$, $\chi_{437}(210,·)$, $\chi_{437}(341,·)$, $\chi_{437}(343,·)$, $\chi_{437}(94,·)$, $\chi_{437}(96,·)$, $\chi_{437}(227,·)$, $\chi_{437}(229,·)$, $\chi_{437}(360,·)$, $\chi_{437}(362,·)$, $\chi_{437}(113,·)$, $\chi_{437}(246,·)$, $\chi_{437}(248,·)$, $\chi_{437}(379,·)$, $\chi_{437}(381,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{114426145}a^{23}+\frac{1}{5}a^{22}-\frac{1}{5}a^{21}+\frac{1}{5}a^{20}-\frac{1}{5}a^{19}+\frac{1}{5}a^{18}-\frac{1}{5}a^{17}+\frac{1}{5}a^{16}-\frac{1}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1171780}{22885229}$, $\frac{1}{572130725}a^{24}-\frac{1}{572130725}a^{23}-\frac{1}{25}a^{22}-\frac{4}{25}a^{21}+\frac{9}{25}a^{20}+\frac{11}{25}a^{19}-\frac{6}{25}a^{18}+\frac{1}{25}a^{17}+\frac{4}{25}a^{16}-\frac{9}{25}a^{15}-\frac{11}{25}a^{14}+\frac{6}{25}a^{13}-\frac{1}{25}a^{12}-\frac{4}{25}a^{11}+\frac{9}{25}a^{10}+\frac{11}{25}a^{9}-\frac{6}{25}a^{8}+\frac{1}{25}a^{7}+\frac{4}{25}a^{6}-\frac{9}{25}a^{5}-\frac{11}{25}a^{4}+\frac{6}{25}a^{3}-\frac{1}{25}a^{2}+\frac{21713449}{114426145}a+\frac{234356}{22885229}$, $\frac{1}{2860653625}a^{25}-\frac{1}{2860653625}a^{24}-\frac{4}{2860653625}a^{23}-\frac{54}{125}a^{22}+\frac{59}{125}a^{21}-\frac{39}{125}a^{20}-\frac{6}{125}a^{19}-\frac{49}{125}a^{18}-\frac{46}{125}a^{17}+\frac{41}{125}a^{16}-\frac{61}{125}a^{15}-\frac{19}{125}a^{14}-\frac{51}{125}a^{13}+\frac{21}{125}a^{12}-\frac{16}{125}a^{11}+\frac{36}{125}a^{10}+\frac{44}{125}a^{9}+\frac{26}{125}a^{8}+\frac{4}{125}a^{7}-\frac{9}{125}a^{6}-\frac{11}{125}a^{5}+\frac{56}{125}a^{4}-\frac{1}{125}a^{3}+\frac{21713449}{572130725}a^{2}+\frac{234356}{114426145}a-\frac{4389561}{22885229}$, $\frac{1}{14303268125}a^{26}-\frac{1}{14303268125}a^{25}-\frac{4}{14303268125}a^{24}+\frac{9}{14303268125}a^{23}-\frac{191}{625}a^{22}-\frac{164}{625}a^{21}-\frac{131}{625}a^{20}-\frac{299}{625}a^{19}-\frac{296}{625}a^{18}-\frac{84}{625}a^{17}-\frac{311}{625}a^{16}+\frac{106}{625}a^{15}+\frac{199}{625}a^{14}-\frac{104}{625}a^{13}-\frac{266}{625}a^{12}+\frac{161}{625}a^{11}-\frac{81}{625}a^{10}-\frac{99}{625}a^{9}-\frac{121}{625}a^{8}-\frac{9}{625}a^{7}-\frac{11}{625}a^{6}+\frac{56}{625}a^{5}-\frac{1}{625}a^{4}+\frac{21713449}{2860653625}a^{3}+\frac{234356}{572130725}a^{2}-\frac{4389561}{114426145}a+\frac{831041}{22885229}$, $\frac{1}{71516340625}a^{27}-\frac{1}{71516340625}a^{26}-\frac{4}{71516340625}a^{25}+\frac{9}{71516340625}a^{24}+\frac{11}{71516340625}a^{23}+\frac{461}{3125}a^{22}+\frac{494}{3125}a^{21}+\frac{326}{3125}a^{20}+\frac{329}{3125}a^{19}+\frac{1166}{3125}a^{18}+\frac{314}{3125}a^{17}+\frac{106}{3125}a^{16}+\frac{1449}{3125}a^{15}+\frac{1146}{3125}a^{14}+\frac{984}{3125}a^{13}-\frac{464}{3125}a^{12}-\frac{1331}{3125}a^{11}+\frac{526}{3125}a^{10}-\frac{121}{3125}a^{9}+\frac{616}{3125}a^{8}-\frac{11}{3125}a^{7}+\frac{56}{3125}a^{6}-\frac{1}{3125}a^{5}+\frac{21713449}{14303268125}a^{4}+\frac{234356}{2860653625}a^{3}-\frac{4389561}{572130725}a^{2}+\frac{831041}{114426145}a+\frac{711704}{22885229}$, $\frac{1}{357581703125}a^{28}-\frac{1}{357581703125}a^{27}-\frac{4}{357581703125}a^{26}+\frac{9}{357581703125}a^{25}+\frac{11}{357581703125}a^{24}-\frac{56}{357581703125}a^{23}+\frac{6744}{15625}a^{22}+\frac{6576}{15625}a^{21}+\frac{6579}{15625}a^{20}+\frac{7416}{15625}a^{19}+\frac{6564}{15625}a^{18}+\frac{3231}{15625}a^{17}-\frac{4801}{15625}a^{16}+\frac{4271}{15625}a^{15}+\frac{4109}{15625}a^{14}+\frac{5786}{15625}a^{13}+\frac{4919}{15625}a^{12}-\frac{2599}{15625}a^{11}-\frac{6371}{15625}a^{10}+\frac{3741}{15625}a^{9}-\frac{3136}{15625}a^{8}+\frac{56}{15625}a^{7}-\frac{1}{15625}a^{6}+\frac{21713449}{71516340625}a^{5}+\frac{234356}{14303268125}a^{4}-\frac{4389561}{2860653625}a^{3}+\frac{831041}{572130725}a^{2}+\frac{711704}{114426145}a-\frac{308549}{22885229}$, $\frac{1}{1787908515625}a^{29}-\frac{1}{1787908515625}a^{28}-\frac{4}{1787908515625}a^{27}+\frac{9}{1787908515625}a^{26}+\frac{11}{1787908515625}a^{25}-\frac{56}{1787908515625}a^{24}+\frac{1}{1787908515625}a^{23}-\frac{9049}{78125}a^{22}-\frac{24671}{78125}a^{21}-\frac{8209}{78125}a^{20}-\frac{24686}{78125}a^{19}-\frac{12394}{78125}a^{18}-\frac{20426}{78125}a^{17}+\frac{4271}{78125}a^{16}+\frac{19734}{78125}a^{15}+\frac{37036}{78125}a^{14}+\frac{20544}{78125}a^{13}+\frac{28651}{78125}a^{12}+\frac{24879}{78125}a^{11}-\frac{11884}{78125}a^{10}-\frac{34386}{78125}a^{9}+\frac{15681}{78125}a^{8}-\frac{1}{78125}a^{7}+\frac{21713449}{357581703125}a^{6}+\frac{234356}{71516340625}a^{5}-\frac{4389561}{14303268125}a^{4}+\frac{831041}{2860653625}a^{3}+\frac{711704}{572130725}a^{2}-\frac{308549}{114426145}a-\frac{80631}{22885229}$, $\frac{1}{8939542578125}a^{30}-\frac{1}{8939542578125}a^{29}-\frac{4}{8939542578125}a^{28}+\frac{9}{8939542578125}a^{27}+\frac{11}{8939542578125}a^{26}-\frac{56}{8939542578125}a^{25}+\frac{1}{8939542578125}a^{24}+\frac{279}{8939542578125}a^{23}+\frac{53454}{390625}a^{22}-\frac{8209}{390625}a^{21}+\frac{131564}{390625}a^{20}-\frac{90519}{390625}a^{19}-\frac{176676}{390625}a^{18}-\frac{151979}{390625}a^{17}-\frac{136516}{390625}a^{16}+\frac{115161}{390625}a^{15}+\frac{176794}{390625}a^{14}+\frac{28651}{390625}a^{13}-\frac{131371}{390625}a^{12}-\frac{11884}{390625}a^{11}-\frac{112511}{390625}a^{10}+\frac{171931}{390625}a^{9}-\frac{1}{390625}a^{8}+\frac{21713449}{1787908515625}a^{7}+\frac{234356}{357581703125}a^{6}-\frac{4389561}{71516340625}a^{5}+\frac{831041}{14303268125}a^{4}+\frac{711704}{2860653625}a^{3}-\frac{308549}{572130725}a^{2}-\frac{80631}{114426145}a+\frac{77836}{22885229}$, $\frac{1}{44697712890625}a^{31}-\frac{1}{44697712890625}a^{30}-\frac{4}{44697712890625}a^{29}+\frac{9}{44697712890625}a^{28}+\frac{11}{44697712890625}a^{27}-\frac{56}{44697712890625}a^{26}+\frac{1}{44697712890625}a^{25}+\frac{279}{44697712890625}a^{24}-\frac{284}{44697712890625}a^{23}+\frac{382416}{1953125}a^{22}-\frac{649686}{1953125}a^{21}+\frac{690731}{1953125}a^{20}+\frac{604574}{1953125}a^{19}-\frac{151979}{1953125}a^{18}-\frac{917766}{1953125}a^{17}-\frac{275464}{1953125}a^{16}+\frac{958044}{1953125}a^{15}+\frac{419276}{1953125}a^{14}+\frac{649879}{1953125}a^{13}-\frac{793134}{1953125}a^{12}-\frac{503136}{1953125}a^{11}+\frac{562556}{1953125}a^{10}-\frac{1}{1953125}a^{9}+\frac{21713449}{8939542578125}a^{8}+\frac{234356}{1787908515625}a^{7}-\frac{4389561}{357581703125}a^{6}+\frac{831041}{71516340625}a^{5}+\frac{711704}{14303268125}a^{4}-\frac{308549}{2860653625}a^{3}-\frac{80631}{572130725}a^{2}+\frac{77836}{114426145}a+\frac{559}{22885229}$, $\frac{1}{223488564453125}a^{32}-\frac{1}{223488564453125}a^{31}-\frac{4}{223488564453125}a^{30}+\frac{9}{223488564453125}a^{29}+\frac{11}{223488564453125}a^{28}-\frac{56}{223488564453125}a^{27}+\frac{1}{223488564453125}a^{26}+\frac{279}{223488564453125}a^{25}-\frac{284}{223488564453125}a^{24}-\frac{1111}{223488564453125}a^{23}-\frac{649686}{9765625}a^{22}-\frac{1262394}{9765625}a^{21}+\frac{4510824}{9765625}a^{20}+\frac{1801146}{9765625}a^{19}-\frac{4824016}{9765625}a^{18}-\frac{4181714}{9765625}a^{17}-\frac{995081}{9765625}a^{16}+\frac{2372401}{9765625}a^{15}+\frac{2603004}{9765625}a^{14}-\frac{4699384}{9765625}a^{13}+\frac{1449989}{9765625}a^{12}+\frac{2515681}{9765625}a^{11}-\frac{1}{9765625}a^{10}+\frac{21713449}{44697712890625}a^{9}+\frac{234356}{8939542578125}a^{8}-\frac{4389561}{1787908515625}a^{7}+\frac{831041}{357581703125}a^{6}+\frac{711704}{71516340625}a^{5}-\frac{308549}{14303268125}a^{4}-\frac{80631}{2860653625}a^{3}+\frac{77836}{572130725}a^{2}+\frac{559}{114426145}a-\frac{15679}{22885229}$, $\frac{1}{11\!\cdots\!25}a^{33}-\frac{1}{11\!\cdots\!25}a^{32}-\frac{4}{11\!\cdots\!25}a^{31}+\frac{9}{11\!\cdots\!25}a^{30}+\frac{11}{11\!\cdots\!25}a^{29}-\frac{56}{11\!\cdots\!25}a^{28}+\frac{1}{11\!\cdots\!25}a^{27}+\frac{279}{11\!\cdots\!25}a^{26}-\frac{284}{11\!\cdots\!25}a^{25}-\frac{1111}{11\!\cdots\!25}a^{24}+\frac{2531}{11\!\cdots\!25}a^{23}+\frac{8503231}{48828125}a^{22}-\frac{5254801}{48828125}a^{21}+\frac{11566771}{48828125}a^{20}+\frac{14707234}{48828125}a^{19}-\frac{23712964}{48828125}a^{18}-\frac{995081}{48828125}a^{17}+\frac{21903651}{48828125}a^{16}-\frac{16928246}{48828125}a^{15}+\frac{5066241}{48828125}a^{14}-\frac{18081261}{48828125}a^{13}-\frac{7249944}{48828125}a^{12}-\frac{1}{48828125}a^{11}+\frac{21713449}{223488564453125}a^{10}+\frac{234356}{44697712890625}a^{9}-\frac{4389561}{8939542578125}a^{8}+\frac{831041}{1787908515625}a^{7}+\frac{711704}{357581703125}a^{6}-\frac{308549}{71516340625}a^{5}-\frac{80631}{14303268125}a^{4}+\frac{77836}{2860653625}a^{3}+\frac{559}{572130725}a^{2}-\frac{15679}{114426145}a+\frac{3024}{22885229}$, $\frac{1}{55\!\cdots\!25}a^{34}-\frac{1}{55\!\cdots\!25}a^{33}-\frac{4}{55\!\cdots\!25}a^{32}+\frac{9}{55\!\cdots\!25}a^{31}+\frac{11}{55\!\cdots\!25}a^{30}-\frac{56}{55\!\cdots\!25}a^{29}+\frac{1}{55\!\cdots\!25}a^{28}+\frac{279}{55\!\cdots\!25}a^{27}-\frac{284}{55\!\cdots\!25}a^{26}-\frac{1111}{55\!\cdots\!25}a^{25}+\frac{2531}{55\!\cdots\!25}a^{24}+\frac{3024}{55\!\cdots\!25}a^{23}-\frac{5254801}{244140625}a^{22}-\frac{37261354}{244140625}a^{21}+\frac{63535359}{244140625}a^{20}-\frac{121369214}{244140625}a^{19}+\frac{47833044}{244140625}a^{18}+\frac{70731776}{244140625}a^{17}-\frac{65756371}{244140625}a^{16}-\frac{43761884}{244140625}a^{15}-\frac{115737511}{244140625}a^{14}+\frac{90406306}{244140625}a^{13}-\frac{1}{244140625}a^{12}+\frac{21713449}{11\!\cdots\!25}a^{11}+\frac{234356}{223488564453125}a^{10}-\frac{4389561}{44697712890625}a^{9}+\frac{831041}{8939542578125}a^{8}+\frac{711704}{1787908515625}a^{7}-\frac{308549}{357581703125}a^{6}-\frac{80631}{71516340625}a^{5}+\frac{77836}{14303268125}a^{4}+\frac{559}{2860653625}a^{3}-\frac{15679}{572130725}a^{2}+\frac{3024}{114426145}a+\frac{2531}{22885229}$, $\frac{1}{27\!\cdots\!25}a^{35}-\frac{1}{27\!\cdots\!25}a^{34}-\frac{4}{27\!\cdots\!25}a^{33}+\frac{9}{27\!\cdots\!25}a^{32}+\frac{11}{27\!\cdots\!25}a^{31}-\frac{56}{27\!\cdots\!25}a^{30}+\frac{1}{27\!\cdots\!25}a^{29}+\frac{279}{27\!\cdots\!25}a^{28}-\frac{284}{27\!\cdots\!25}a^{27}-\frac{1111}{27\!\cdots\!25}a^{26}+\frac{2531}{27\!\cdots\!25}a^{25}+\frac{3024}{27\!\cdots\!25}a^{24}-\frac{15679}{27\!\cdots\!25}a^{23}-\frac{281401979}{1220703125}a^{22}+\frac{307675984}{1220703125}a^{21}-\frac{121369214}{1220703125}a^{20}-\frac{196307581}{1220703125}a^{19}-\frac{417549474}{1220703125}a^{18}+\frac{178384254}{1220703125}a^{17}-\frac{532043134}{1220703125}a^{16}-\frac{359878136}{1220703125}a^{15}+\frac{578687556}{1220703125}a^{14}-\frac{1}{1220703125}a^{13}+\frac{21713449}{55\!\cdots\!25}a^{12}+\frac{234356}{11\!\cdots\!25}a^{11}-\frac{4389561}{223488564453125}a^{10}+\frac{831041}{44697712890625}a^{9}+\frac{711704}{8939542578125}a^{8}-\frac{308549}{1787908515625}a^{7}-\frac{80631}{357581703125}a^{6}+\frac{77836}{71516340625}a^{5}+\frac{559}{14303268125}a^{4}-\frac{15679}{2860653625}a^{3}+\frac{3024}{572130725}a^{2}+\frac{2531}{114426145}a-\frac{1111}{22885229}$, $\frac{1}{13\!\cdots\!25}a^{36}-\frac{1}{13\!\cdots\!25}a^{35}-\frac{4}{13\!\cdots\!25}a^{34}+\frac{9}{13\!\cdots\!25}a^{33}+\frac{11}{13\!\cdots\!25}a^{32}-\frac{56}{13\!\cdots\!25}a^{31}+\frac{1}{13\!\cdots\!25}a^{30}+\frac{279}{13\!\cdots\!25}a^{29}-\frac{284}{13\!\cdots\!25}a^{28}-\frac{1111}{13\!\cdots\!25}a^{27}+\frac{2531}{13\!\cdots\!25}a^{26}+\frac{3024}{13\!\cdots\!25}a^{25}-\frac{15679}{13\!\cdots\!25}a^{24}+\frac{559}{13\!\cdots\!25}a^{23}+\frac{307675984}{6103515625}a^{22}+\frac{1099333911}{6103515625}a^{21}-\frac{2637713831}{6103515625}a^{20}-\frac{2858955724}{6103515625}a^{19}-\frac{2263021996}{6103515625}a^{18}-\frac{1752746259}{6103515625}a^{17}+\frac{860824989}{6103515625}a^{16}+\frac{1799390681}{6103515625}a^{15}-\frac{1}{6103515625}a^{14}+\frac{21713449}{27\!\cdots\!25}a^{13}+\frac{234356}{55\!\cdots\!25}a^{12}-\frac{4389561}{11\!\cdots\!25}a^{11}+\frac{831041}{223488564453125}a^{10}+\frac{711704}{44697712890625}a^{9}-\frac{308549}{8939542578125}a^{8}-\frac{80631}{1787908515625}a^{7}+\frac{77836}{357581703125}a^{6}+\frac{559}{71516340625}a^{5}-\frac{15679}{14303268125}a^{4}+\frac{3024}{2860653625}a^{3}+\frac{2531}{572130725}a^{2}-\frac{1111}{114426145}a-\frac{284}{22885229}$, $\frac{1}{69\!\cdots\!25}a^{37}-\frac{1}{69\!\cdots\!25}a^{36}-\frac{4}{69\!\cdots\!25}a^{35}+\frac{9}{69\!\cdots\!25}a^{34}+\frac{11}{69\!\cdots\!25}a^{33}-\frac{56}{69\!\cdots\!25}a^{32}+\frac{1}{69\!\cdots\!25}a^{31}+\frac{279}{69\!\cdots\!25}a^{30}-\frac{284}{69\!\cdots\!25}a^{29}-\frac{1111}{69\!\cdots\!25}a^{28}+\frac{2531}{69\!\cdots\!25}a^{27}+\frac{3024}{69\!\cdots\!25}a^{26}-\frac{15679}{69\!\cdots\!25}a^{25}+\frac{559}{69\!\cdots\!25}a^{24}+\frac{77836}{69\!\cdots\!25}a^{23}+\frac{1099333911}{30517578125}a^{22}-\frac{2637713831}{30517578125}a^{21}-\frac{2858955724}{30517578125}a^{20}-\frac{14470053246}{30517578125}a^{19}-\frac{1752746259}{30517578125}a^{18}+\frac{13067856239}{30517578125}a^{17}-\frac{4304124944}{30517578125}a^{16}-\frac{1}{30517578125}a^{15}+\frac{21713449}{13\!\cdots\!25}a^{14}+\frac{234356}{27\!\cdots\!25}a^{13}-\frac{4389561}{55\!\cdots\!25}a^{12}+\frac{831041}{11\!\cdots\!25}a^{11}+\frac{711704}{223488564453125}a^{10}-\frac{308549}{44697712890625}a^{9}-\frac{80631}{8939542578125}a^{8}+\frac{77836}{1787908515625}a^{7}+\frac{559}{357581703125}a^{6}-\frac{15679}{71516340625}a^{5}+\frac{3024}{14303268125}a^{4}+\frac{2531}{2860653625}a^{3}-\frac{1111}{572130725}a^{2}-\frac{284}{114426145}a+\frac{279}{22885229}$, $\frac{1}{34\!\cdots\!25}a^{38}-\frac{1}{34\!\cdots\!25}a^{37}-\frac{4}{34\!\cdots\!25}a^{36}+\frac{9}{34\!\cdots\!25}a^{35}+\frac{11}{34\!\cdots\!25}a^{34}-\frac{56}{34\!\cdots\!25}a^{33}+\frac{1}{34\!\cdots\!25}a^{32}+\frac{279}{34\!\cdots\!25}a^{31}-\frac{284}{34\!\cdots\!25}a^{30}-\frac{1111}{34\!\cdots\!25}a^{29}+\frac{2531}{34\!\cdots\!25}a^{28}+\frac{3024}{34\!\cdots\!25}a^{27}-\frac{15679}{34\!\cdots\!25}a^{26}+\frac{559}{34\!\cdots\!25}a^{25}+\frac{77836}{34\!\cdots\!25}a^{24}-\frac{80631}{34\!\cdots\!25}a^{23}-\frac{2637713831}{152587890625}a^{22}-\frac{2858955724}{152587890625}a^{21}+\frac{16047524879}{152587890625}a^{20}-\frac{1752746259}{152587890625}a^{19}+\frac{74103012489}{152587890625}a^{18}-\frac{65339281194}{152587890625}a^{17}-\frac{1}{152587890625}a^{16}+\frac{21713449}{69\!\cdots\!25}a^{15}+\frac{234356}{13\!\cdots\!25}a^{14}-\frac{4389561}{27\!\cdots\!25}a^{13}+\frac{831041}{55\!\cdots\!25}a^{12}+\frac{711704}{11\!\cdots\!25}a^{11}-\frac{308549}{223488564453125}a^{10}-\frac{80631}{44697712890625}a^{9}+\frac{77836}{8939542578125}a^{8}+\frac{559}{1787908515625}a^{7}-\frac{15679}{357581703125}a^{6}+\frac{3024}{71516340625}a^{5}+\frac{2531}{14303268125}a^{4}-\frac{1111}{2860653625}a^{3}-\frac{284}{572130725}a^{2}+\frac{279}{114426145}a+\frac{1}{22885229}$, $\frac{1}{17\!\cdots\!25}a^{39}-\frac{1}{17\!\cdots\!25}a^{38}-\frac{4}{17\!\cdots\!25}a^{37}+\frac{9}{17\!\cdots\!25}a^{36}+\frac{11}{17\!\cdots\!25}a^{35}-\frac{56}{17\!\cdots\!25}a^{34}+\frac{1}{17\!\cdots\!25}a^{33}+\frac{279}{17\!\cdots\!25}a^{32}-\frac{284}{17\!\cdots\!25}a^{31}-\frac{1111}{17\!\cdots\!25}a^{30}+\frac{2531}{17\!\cdots\!25}a^{29}+\frac{3024}{17\!\cdots\!25}a^{28}-\frac{15679}{17\!\cdots\!25}a^{27}+\frac{559}{17\!\cdots\!25}a^{26}+\frac{77836}{17\!\cdots\!25}a^{25}-\frac{80631}{17\!\cdots\!25}a^{24}-\frac{308549}{17\!\cdots\!25}a^{23}+\frac{302316825526}{762939453125}a^{22}-\frac{289128256371}{762939453125}a^{21}+\frac{303423034991}{762939453125}a^{20}+\frac{379278793739}{762939453125}a^{19}-\frac{370515062444}{762939453125}a^{18}-\frac{1}{762939453125}a^{17}+\frac{21713449}{34\!\cdots\!25}a^{16}+\frac{234356}{69\!\cdots\!25}a^{15}-\frac{4389561}{13\!\cdots\!25}a^{14}+\frac{831041}{27\!\cdots\!25}a^{13}+\frac{711704}{55\!\cdots\!25}a^{12}-\frac{308549}{11\!\cdots\!25}a^{11}-\frac{80631}{223488564453125}a^{10}+\frac{77836}{44697712890625}a^{9}+\frac{559}{8939542578125}a^{8}-\frac{15679}{1787908515625}a^{7}+\frac{3024}{357581703125}a^{6}+\frac{2531}{71516340625}a^{5}-\frac{1111}{14303268125}a^{4}-\frac{284}{2860653625}a^{3}+\frac{279}{572130725}a^{2}+\frac{1}{114426145}a-\frac{56}{22885229}$, $\frac{1}{87\!\cdots\!25}a^{40}-\frac{1}{87\!\cdots\!25}a^{39}-\frac{4}{87\!\cdots\!25}a^{38}+\frac{9}{87\!\cdots\!25}a^{37}+\frac{11}{87\!\cdots\!25}a^{36}-\frac{56}{87\!\cdots\!25}a^{35}+\frac{1}{87\!\cdots\!25}a^{34}+\frac{279}{87\!\cdots\!25}a^{33}-\frac{284}{87\!\cdots\!25}a^{32}-\frac{1111}{87\!\cdots\!25}a^{31}+\frac{2531}{87\!\cdots\!25}a^{30}+\frac{3024}{87\!\cdots\!25}a^{29}-\frac{15679}{87\!\cdots\!25}a^{28}+\frac{559}{87\!\cdots\!25}a^{27}+\frac{77836}{87\!\cdots\!25}a^{26}-\frac{80631}{87\!\cdots\!25}a^{25}-\frac{308549}{87\!\cdots\!25}a^{24}+\frac{711704}{87\!\cdots\!25}a^{23}+\frac{1236750649879}{3814697265625}a^{22}+\frac{1066362488116}{3814697265625}a^{21}+\frac{379278793739}{3814697265625}a^{20}-\frac{1896393968694}{3814697265625}a^{19}-\frac{1}{3814697265625}a^{18}+\frac{21713449}{17\!\cdots\!25}a^{17}+\frac{234356}{34\!\cdots\!25}a^{16}-\frac{4389561}{69\!\cdots\!25}a^{15}+\frac{831041}{13\!\cdots\!25}a^{14}+\frac{711704}{27\!\cdots\!25}a^{13}-\frac{308549}{55\!\cdots\!25}a^{12}-\frac{80631}{11\!\cdots\!25}a^{11}+\frac{77836}{223488564453125}a^{10}+\frac{559}{44697712890625}a^{9}-\frac{15679}{8939542578125}a^{8}+\frac{3024}{1787908515625}a^{7}+\frac{2531}{357581703125}a^{6}-\frac{1111}{71516340625}a^{5}-\frac{284}{14303268125}a^{4}+\frac{279}{2860653625}a^{3}+\frac{1}{572130725}a^{2}-\frac{56}{114426145}a+\frac{11}{22885229}$, $\frac{1}{43\!\cdots\!25}a^{41}-\frac{1}{43\!\cdots\!25}a^{40}-\frac{4}{43\!\cdots\!25}a^{39}+\frac{9}{43\!\cdots\!25}a^{38}+\frac{11}{43\!\cdots\!25}a^{37}-\frac{56}{43\!\cdots\!25}a^{36}+\frac{1}{43\!\cdots\!25}a^{35}+\frac{279}{43\!\cdots\!25}a^{34}-\frac{284}{43\!\cdots\!25}a^{33}-\frac{1111}{43\!\cdots\!25}a^{32}+\frac{2531}{43\!\cdots\!25}a^{31}+\frac{3024}{43\!\cdots\!25}a^{30}-\frac{15679}{43\!\cdots\!25}a^{29}+\frac{559}{43\!\cdots\!25}a^{28}+\frac{77836}{43\!\cdots\!25}a^{27}-\frac{80631}{43\!\cdots\!25}a^{26}-\frac{308549}{43\!\cdots\!25}a^{25}+\frac{711704}{43\!\cdots\!25}a^{24}+\frac{831041}{43\!\cdots\!25}a^{23}-\frac{2748334777509}{19073486328125}a^{22}-\frac{3435418471886}{19073486328125}a^{21}-\frac{1896393968694}{19073486328125}a^{20}-\frac{1}{19073486328125}a^{19}+\frac{21713449}{87\!\cdots\!25}a^{18}+\frac{234356}{17\!\cdots\!25}a^{17}-\frac{4389561}{34\!\cdots\!25}a^{16}+\frac{831041}{69\!\cdots\!25}a^{15}+\frac{711704}{13\!\cdots\!25}a^{14}-\frac{308549}{27\!\cdots\!25}a^{13}-\frac{80631}{55\!\cdots\!25}a^{12}+\frac{77836}{11\!\cdots\!25}a^{11}+\frac{559}{223488564453125}a^{10}-\frac{15679}{44697712890625}a^{9}+\frac{3024}{8939542578125}a^{8}+\frac{2531}{1787908515625}a^{7}-\frac{1111}{357581703125}a^{6}-\frac{284}{71516340625}a^{5}+\frac{279}{14303268125}a^{4}+\frac{1}{2860653625}a^{3}-\frac{56}{572130725}a^{2}+\frac{11}{114426145}a+\frac{9}{22885229}$, $\frac{1}{21\!\cdots\!25}a^{42}-\frac{1}{21\!\cdots\!25}a^{41}-\frac{4}{21\!\cdots\!25}a^{40}+\frac{9}{21\!\cdots\!25}a^{39}+\frac{11}{21\!\cdots\!25}a^{38}-\frac{56}{21\!\cdots\!25}a^{37}+\frac{1}{21\!\cdots\!25}a^{36}+\frac{279}{21\!\cdots\!25}a^{35}-\frac{284}{21\!\cdots\!25}a^{34}-\frac{1111}{21\!\cdots\!25}a^{33}+\frac{2531}{21\!\cdots\!25}a^{32}+\frac{3024}{21\!\cdots\!25}a^{31}-\frac{15679}{21\!\cdots\!25}a^{30}+\frac{559}{21\!\cdots\!25}a^{29}+\frac{77836}{21\!\cdots\!25}a^{28}-\frac{80631}{21\!\cdots\!25}a^{27}-\frac{308549}{21\!\cdots\!25}a^{26}+\frac{711704}{21\!\cdots\!25}a^{25}+\frac{831041}{21\!\cdots\!25}a^{24}-\frac{4389561}{21\!\cdots\!25}a^{23}-\frac{3435418471886}{95367431640625}a^{22}+\frac{17177092359431}{95367431640625}a^{21}-\frac{1}{95367431640625}a^{20}+\frac{21713449}{43\!\cdots\!25}a^{19}+\frac{234356}{87\!\cdots\!25}a^{18}-\frac{4389561}{17\!\cdots\!25}a^{17}+\frac{831041}{34\!\cdots\!25}a^{16}+\frac{711704}{69\!\cdots\!25}a^{15}-\frac{308549}{13\!\cdots\!25}a^{14}-\frac{80631}{27\!\cdots\!25}a^{13}+\frac{77836}{55\!\cdots\!25}a^{12}+\frac{559}{11\!\cdots\!25}a^{11}-\frac{15679}{223488564453125}a^{10}+\frac{3024}{44697712890625}a^{9}+\frac{2531}{8939542578125}a^{8}-\frac{1111}{1787908515625}a^{7}-\frac{284}{357581703125}a^{6}+\frac{279}{71516340625}a^{5}+\frac{1}{14303268125}a^{4}-\frac{56}{2860653625}a^{3}+\frac{11}{572130725}a^{2}+\frac{9}{114426145}a-\frac{4}{22885229}$, $\frac{1}{10\!\cdots\!25}a^{43}-\frac{1}{10\!\cdots\!25}a^{42}-\frac{4}{10\!\cdots\!25}a^{41}+\frac{9}{10\!\cdots\!25}a^{40}+\frac{11}{10\!\cdots\!25}a^{39}-\frac{56}{10\!\cdots\!25}a^{38}+\frac{1}{10\!\cdots\!25}a^{37}+\frac{279}{10\!\cdots\!25}a^{36}-\frac{284}{10\!\cdots\!25}a^{35}-\frac{1111}{10\!\cdots\!25}a^{34}+\frac{2531}{10\!\cdots\!25}a^{33}+\frac{3024}{10\!\cdots\!25}a^{32}-\frac{15679}{10\!\cdots\!25}a^{31}+\frac{559}{10\!\cdots\!25}a^{30}+\frac{77836}{10\!\cdots\!25}a^{29}-\frac{80631}{10\!\cdots\!25}a^{28}-\frac{308549}{10\!\cdots\!25}a^{27}+\frac{711704}{10\!\cdots\!25}a^{26}+\frac{831041}{10\!\cdots\!25}a^{25}-\frac{4389561}{10\!\cdots\!25}a^{24}+\frac{234356}{10\!\cdots\!25}a^{23}+\frac{17177092359431}{476837158203125}a^{22}-\frac{1}{476837158203125}a^{21}+\frac{21713449}{21\!\cdots\!25}a^{20}+\frac{234356}{43\!\cdots\!25}a^{19}-\frac{4389561}{87\!\cdots\!25}a^{18}+\frac{831041}{17\!\cdots\!25}a^{17}+\frac{711704}{34\!\cdots\!25}a^{16}-\frac{308549}{69\!\cdots\!25}a^{15}-\frac{80631}{13\!\cdots\!25}a^{14}+\frac{77836}{27\!\cdots\!25}a^{13}+\frac{559}{55\!\cdots\!25}a^{12}-\frac{15679}{11\!\cdots\!25}a^{11}+\frac{3024}{223488564453125}a^{10}+\frac{2531}{44697712890625}a^{9}-\frac{1111}{8939542578125}a^{8}-\frac{284}{1787908515625}a^{7}+\frac{279}{357581703125}a^{6}+\frac{1}{71516340625}a^{5}-\frac{56}{14303268125}a^{4}+\frac{11}{2860653625}a^{3}+\frac{9}{572130725}a^{2}-\frac{4}{114426145}a-\frac{1}{22885229}$ 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, a^20, a^21, a^22, 1/114426145*a^23 + 1/5*a^22 - 1/5*a^21 + 1/5*a^20 - 1/5*a^19 + 1/5*a^18 - 1/5*a^17 + 1/5*a^16 - 1/5*a^15 + 1/5*a^14 - 1/5*a^13 + 1/5*a^12 - 1/5*a^11 + 1/5*a^10 - 1/5*a^9 + 1/5*a^8 - 1/5*a^7 + 1/5*a^6 - 1/5*a^5 + 1/5*a^4 - 1/5*a^3 + 1/5*a^2 - 1/5*a - 1171780/22885229, 1/572130725*a^24 - 1/572130725*a^23 - 1/25*a^22 - 4/25*a^21 + 9/25*a^20 + 11/25*a^19 - 6/25*a^18 + 1/25*a^17 + 4/25*a^16 - 9/25*a^15 - 11/25*a^14 + 6/25*a^13 - 1/25*a^12 - 4/25*a^11 + 9/25*a^10 + 11/25*a^9 - 6/25*a^8 + 1/25*a^7 + 4/25*a^6 - 9/25*a^5 - 11/25*a^4 + 6/25*a^3 - 1/25*a^2 + 21713449/114426145*a + 234356/22885229, 1/2860653625*a^25 - 1/2860653625*a^24 - 4/2860653625*a^23 - 54/125*a^22 + 59/125*a^21 - 39/125*a^20 - 6/125*a^19 - 49/125*a^18 - 46/125*a^17 + 41/125*a^16 - 61/125*a^15 - 19/125*a^14 - 51/125*a^13 + 21/125*a^12 - 16/125*a^11 + 36/125*a^10 + 44/125*a^9 + 26/125*a^8 + 4/125*a^7 - 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11/436501102447509765625*a^37 - 56/436501102447509765625*a^36 + 1/436501102447509765625*a^35 + 279/436501102447509765625*a^34 - 284/436501102447509765625*a^33 - 1111/436501102447509765625*a^32 + 2531/436501102447509765625*a^31 + 3024/436501102447509765625*a^30 - 15679/436501102447509765625*a^29 + 559/436501102447509765625*a^28 + 77836/436501102447509765625*a^27 - 80631/436501102447509765625*a^26 - 308549/436501102447509765625*a^25 + 711704/436501102447509765625*a^24 + 831041/436501102447509765625*a^23 - 2748334777509/19073486328125*a^22 - 3435418471886/19073486328125*a^21 - 1896393968694/19073486328125*a^20 - 1/19073486328125*a^19 + 21713449/87300220489501953125*a^18 + 234356/17460044097900390625*a^17 - 4389561/3492008819580078125*a^16 + 831041/698401763916015625*a^15 + 711704/139680352783203125*a^14 - 308549/27936070556640625*a^13 - 80631/5587214111328125*a^12 + 77836/1117442822265625*a^11 + 559/223488564453125*a^10 - 15679/44697712890625*a^9 + 3024/8939542578125*a^8 + 2531/1787908515625*a^7 - 1111/357581703125*a^6 - 284/71516340625*a^5 + 279/14303268125*a^4 + 1/2860653625*a^3 - 56/572130725*a^2 + 11/114426145*a + 9/22885229, 1/2182505512237548828125*a^42 - 1/2182505512237548828125*a^41 - 4/2182505512237548828125*a^40 + 9/2182505512237548828125*a^39 + 11/2182505512237548828125*a^38 - 56/2182505512237548828125*a^37 + 1/2182505512237548828125*a^36 + 279/2182505512237548828125*a^35 - 284/2182505512237548828125*a^34 - 1111/2182505512237548828125*a^33 + 2531/2182505512237548828125*a^32 + 3024/2182505512237548828125*a^31 - 15679/2182505512237548828125*a^30 + 559/2182505512237548828125*a^29 + 77836/2182505512237548828125*a^28 - 80631/2182505512237548828125*a^27 - 308549/2182505512237548828125*a^26 + 711704/2182505512237548828125*a^25 + 831041/2182505512237548828125*a^24 - 4389561/2182505512237548828125*a^23 - 3435418471886/95367431640625*a^22 + 17177092359431/95367431640625*a^21 - 1/95367431640625*a^20 + 21713449/436501102447509765625*a^19 + 234356/87300220489501953125*a^18 - 4389561/17460044097900390625*a^17 + 831041/3492008819580078125*a^16 + 711704/698401763916015625*a^15 - 308549/139680352783203125*a^14 - 80631/27936070556640625*a^13 + 77836/5587214111328125*a^12 + 559/1117442822265625*a^11 - 15679/223488564453125*a^10 + 3024/44697712890625*a^9 + 2531/8939542578125*a^8 - 1111/1787908515625*a^7 - 284/357581703125*a^6 + 279/71516340625*a^5 + 1/14303268125*a^4 - 56/2860653625*a^3 + 11/572130725*a^2 + 9/114426145*a - 4/22885229, 1/10912527561187744140625*a^43 - 1/10912527561187744140625*a^42 - 4/10912527561187744140625*a^41 + 9/10912527561187744140625*a^40 + 11/10912527561187744140625*a^39 - 56/10912527561187744140625*a^38 + 1/10912527561187744140625*a^37 + 279/10912527561187744140625*a^36 - 284/10912527561187744140625*a^35 - 1111/10912527561187744140625*a^34 + 2531/10912527561187744140625*a^33 + 3024/10912527561187744140625*a^32 - 15679/10912527561187744140625*a^31 + 559/10912527561187744140625*a^30 + 77836/10912527561187744140625*a^29 - 80631/10912527561187744140625*a^28 - 308549/10912527561187744140625*a^27 + 711704/10912527561187744140625*a^26 + 831041/10912527561187744140625*a^25 - 4389561/10912527561187744140625*a^24 + 234356/10912527561187744140625*a^23 + 17177092359431/476837158203125*a^22 - 1/476837158203125*a^21 + 21713449/2182505512237548828125*a^20 + 234356/436501102447509765625*a^19 - 4389561/87300220489501953125*a^18 + 831041/17460044097900390625*a^17 + 711704/3492008819580078125*a^16 - 308549/698401763916015625*a^15 - 80631/139680352783203125*a^14 + 77836/27936070556640625*a^13 + 559/5587214111328125*a^12 - 15679/1117442822265625*a^11 + 3024/223488564453125*a^10 + 2531/44697712890625*a^9 - 1111/8939542578125*a^8 - 284/1787908515625*a^7 + 279/357581703125*a^6 + 1/71516340625*a^5 - 56/14303268125*a^4 + 11/2860653625*a^3 + 9/572130725*a^2 - 4/114426145*a - 1/22885229

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:	$21$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{831041}{436501102447509765625} a^{42} + \frac{10232276842424}{436501102447509765625} a^{19} $ -(831041)/(436501102447509765625)a^(42) + (10232276842424)/(436501102447509765625)a^(19) (order $46$)	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^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625)

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^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625, 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^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625);

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^44 - x^43 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625);

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

$C_2\times C_{22}$ (as 44T2):

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 44

The 44 conjugacy class representatives for $C_2\times C_{22}$

Character table for $C_2\times C_{22}$

Intermediate fields

$\Q(\sqrt{437}) $, $\Q(\sqrt{-23}) $, $\Q(\sqrt{-19}) $, $\Q(\sqrt{-19}, \sqrt{-23})$, $\Q(\zeta_{23})^+$, 22.22.4598055053342647107748042243736831875581437.1, $\Q(\zeta_{23})$, 22.0.199915437101854222076001836684210081547019.1

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	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	R	R	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/47.1.0.1}{1} }^{44}$	$22^{2}$	$22^{2}$

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
$19$ 19		Deg $44$	$2$	$22$	$22$
$23$ 23	23.22.21.17	$x^{22} + 23$	$22$	$1$	$21$	22T1	$[\ ]_{22}$
$23$ 23	23.22.21.17	$x^{22} + 23$	$22$	$1$	$21$	22T1	$[\ ]_{22}$

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