LMFDB - Number field 46.0.20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879)

gp: K = bnfinit(y^46 - y^45 + 48*y^44 - 48*y^43 + 1082*y^42 - 1082*y^41 + 15229*y^40 - 15229*y^39 + 150119*y^38 - 150119*y^37 + 1102057*y^36 - 1102057*y^35 + 6256453*y^34 - 6256453*y^33 + 28162636*y^32 - 28162636*y^31 + 102306640*y^30 - 102306640*y^29 + 303926300*y^28 - 303926300*y^27 + 746399716*y^26 - 746399716*y^25 + 1530784408*y^24 - 1530784408*y^23 + 2651333968*y^22 - 2651333968*y^21 + 3934135048*y^20 - 3934135048*y^19 + 5100317848*y^18 - 5100317848*y^17 + 5931223093*y^16 - 5931223093*y^15 + 6386880808*y^14 - 6386880808*y^13 + 6574504573*y^12 - 6574504573*y^11 + 6630576043*y^10 - 6630576043*y^9 + 6642169768*y^8 - 6642169768*y^7 + 6643715598*y^6 - 6643715598*y^5 + 6643834508*y^4 - 6643834508*y^3 + 6643838832*y^2 - 6643838832*y + 6643838879, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879)

$ x^{46} - x^{45} + 48 x^{44} - 48 x^{43} + 1082 x^{42} - 1082 x^{41} + 15229 x^{40} - 15229 x^{39} + \cdots + 6643838879 $ x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$46$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 23]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-209\!\cdots\!875$ -20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875 $\medspace = -\,5^{23}\cdot 47^{45}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$96.66$	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:	$5^{1/2}47^{45/46}\approx 96.65690988752236$
Ramified primes:	$5$, $47$ 5, 47	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-235}) $
$\card{ \Gal(K/\Q) }$:	$46$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$235=5\cdot 47$
Dirichlet character group:	$\lbrace$$\chi_{235}(1,·)$, $\chi_{235}(131,·)$, $\chi_{235}(179,·)$, $\chi_{235}(134,·)$, $\chi_{235}(129,·)$, $\chi_{235}(136,·)$, $\chi_{235}(139,·)$, $\chi_{235}(16,·)$, $\chi_{235}(19,·)$, $\chi_{235}(21,·)$, $\chi_{235}(154,·)$, $\chi_{235}(29,·)$, $\chi_{235}(36,·)$, $\chi_{235}(6,·)$, $\chi_{235}(166,·)$, $\chi_{235}(39,·)$, $\chi_{235}(71,·)$, $\chi_{235}(44,·)$, $\chi_{235}(174,·)$, $\chi_{235}(51,·)$, $\chi_{235}(184,·)$, $\chi_{235}(61,·)$, $\chi_{235}(191,·)$, $\chi_{235}(196,·)$, $\chi_{235}(69,·)$, $\chi_{235}(199,·)$, $\chi_{235}(206,·)$, $\chi_{235}(56,·)$, $\chi_{235}(214,·)$, $\chi_{235}(216,·)$, $\chi_{235}(164,·)$, $\chi_{235}(219,·)$, $\chi_{235}(101,·)$, $\chi_{235}(96,·)$, $\chi_{235}(99,·)$, $\chi_{235}(229,·)$, $\chi_{235}(81,·)$, $\chi_{235}(104,·)$, $\chi_{235}(106,·)$, $\chi_{235}(109,·)$, $\chi_{235}(111,·)$, $\chi_{235}(114,·)$, $\chi_{235}(121,·)$, $\chi_{235}(124,·)$, $\chi_{235}(234,·)$, $\chi_{235}(126,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2971215073}a^{24}+\frac{1134903170}{2971215073}a^{23}+\frac{24}{2971215073}a^{22}-\frac{638162747}{2971215073}a^{21}+\frac{252}{2971215073}a^{20}-\frac{439197324}{2971215073}a^{19}+\frac{1520}{2971215073}a^{18}-\frac{720695703}{2971215073}a^{17}+\frac{5814}{2971215073}a^{16}+\frac{548262824}{2971215073}a^{15}+\frac{14688}{2971215073}a^{14}-\frac{701530126}{2971215073}a^{13}+\frac{24752}{2971215073}a^{12}-\frac{1072928428}{2971215073}a^{11}+\frac{27456}{2971215073}a^{10}+\frac{1280757137}{2971215073}a^{9}+\frac{19305}{2971215073}a^{8}-\frac{420031747}{2971215073}a^{7}+\frac{8008}{2971215073}a^{6}+\frac{850394442}{2971215073}a^{5}+\frac{1716}{2971215073}a^{4}+\frac{816494931}{2971215073}a^{3}+\frac{144}{2971215073}a^{2}-\frac{638162747}{2971215073}a+\frac{2}{2971215073}$, $\frac{1}{2971215073}a^{25}+\frac{25}{2971215073}a^{23}-\frac{1134903170}{2971215073}a^{22}+\frac{275}{2971215073}a^{21}-\frac{1198149156}{2971215073}a^{20}+\frac{1750}{2971215073}a^{19}+\frac{502443310}{2971215073}a^{18}+\frac{7125}{2971215073}a^{17}+\frac{1289909577}{2971215073}a^{16}+\frac{19380}{2971215073}a^{15}+\frac{1328483517}{2971215073}a^{14}+\frac{35700}{2971215073}a^{13}+\frac{642322947}{2971215073}a^{12}+\frac{44200}{2971215073}a^{11}+\frac{511792168}{2971215073}a^{10}+\frac{35750}{2971215073}a^{9}+\frac{14219705}{2971215073}a^{8}+\frac{17875}{2971215073}a^{7}-\frac{1478497684}{2971215073}a^{6}+\frac{5005}{2971215073}a^{5}-\frac{531471974}{2971215073}a^{4}+\frac{650}{2971215073}a^{3}-\frac{647390212}{2971215073}a^{2}+\frac{25}{2971215073}a+\frac{701408733}{2971215073}$, $\frac{1}{2971215073}a^{26}+\frac{204668310}{2971215073}a^{23}-\frac{325}{2971215073}a^{22}-\frac{100155846}{2971215073}a^{21}-\frac{4550}{2971215073}a^{20}-\frac{402483882}{2971215073}a^{19}-\frac{30875}{2971215073}a^{18}+\frac{1480011714}{2971215073}a^{17}-\frac{125970}{2971215073}a^{16}-\frac{493226791}{2971215073}a^{15}-\frac{331500}{2971215073}a^{14}+\frac{353285659}{2971215073}a^{13}-\frac{574600}{2971215073}a^{12}+\frac{594067211}{2971215073}a^{11}-\frac{650650}{2971215073}a^{10}+\frac{678657083}{2971215073}a^{9}-\frac{464750}{2971215073}a^{8}+\frac{108650772}{2971215073}a^{7}-\frac{195195}{2971215073}a^{6}-\frac{992827513}{2971215073}a^{5}-\frac{42250}{2971215073}a^{4}-\frac{261257976}{2971215073}a^{3}-\frac{3575}{2971215073}a^{2}-\frac{1171813030}{2971215073}a-\frac{50}{2971215073}$, $\frac{1}{2971215073}a^{27}-\frac{351}{2971215073}a^{23}+\frac{930234860}{2971215073}a^{22}-\frac{5148}{2971215073}a^{21}-\frac{1468241761}{2971215073}a^{20}-\frac{36855}{2971215073}a^{19}-\frac{609451894}{2971215073}a^{18}-\frac{160056}{2971215073}a^{17}+\frac{1022463142}{2971215073}a^{16}-\frac{453492}{2971215073}a^{15}+\frac{1054802255}{2971215073}a^{14}-\frac{859248}{2971215073}a^{13}+\frac{565757556}{2971215073}a^{12}-\frac{1085994}{2971215073}a^{11}-\frac{126759234}{2971215073}a^{10}-\frac{892320}{2971215073}a^{9}+\frac{702973312}{2971215073}a^{8}-\frac{451737}{2971215073}a^{7}+\frac{134066303}{2971215073}a^{6}-\frac{127764}{2971215073}a^{5}-\frac{868699322}{2971215073}a^{4}-\frac{16731}{2971215073}a^{3}-\frac{931898940}{2971215073}a^{2}-\frac{648}{2971215073}a-\frac{409336620}{2971215073}$, $\frac{1}{2971215073}a^{28}+\frac{1138427748}{2971215073}a^{23}+\frac{3276}{2971215073}a^{22}+\frac{348979590}{2971215073}a^{21}+\frac{51597}{2971215073}a^{20}-\frac{264528822}{2971215073}a^{19}+\frac{373464}{2971215073}a^{18}+\frac{611552594}{2971215073}a^{17}+\frac{1587222}{2971215073}a^{16}+\frac{366073734}{2971215073}a^{15}+\frac{4296240}{2971215073}a^{14}+\frac{939534389}{2971215073}a^{13}+\frac{7601958}{2971215073}a^{12}+\frac{619676809}{2971215073}a^{11}+\frac{8744736}{2971215073}a^{10}-\frac{1375962697}{2971215073}a^{9}+\frac{6324318}{2971215073}a^{8}+\frac{1263676756}{2971215073}a^{7}+\frac{2683044}{2971215073}a^{6}+\frac{498242520}{2971215073}a^{5}+\frac{585585}{2971215073}a^{4}+\frac{421174833}{2971215073}a^{3}+\frac{49896}{2971215073}a^{2}+\frac{1407884731}{2971215073}a+\frac{702}{2971215073}$, $\frac{1}{2971215073}a^{29}+\frac{3654}{2971215073}a^{23}-\frac{232350705}{2971215073}a^{22}+\frac{60291}{2971215073}a^{21}+\frac{1059540763}{2971215073}a^{20}+\frac{460404}{2971215073}a^{19}-\frac{551451880}{2971215073}a^{18}+\frac{2082780}{2971215073}a^{17}+\frac{1414329506}{2971215073}a^{16}+\frac{6069816}{2971215073}a^{15}-\frac{1260012464}{2971215073}a^{14}+\frac{11740302}{2971215073}a^{13}+\frac{1259810645}{2971215073}a^{12}+\frac{15073968}{2971215073}a^{11}-\frac{865643825}{2971215073}a^{10}+\frac{12540528}{2971215073}a^{9}-\frac{977318476}{2971215073}a^{8}+\frac{6412770}{2971215073}a^{7}-\frac{343319500}{2971215073}a^{6}+\frac{1828827}{2971215073}a^{5}-\frac{1032537774}{2971215073}a^{4}+\frac{241164}{2971215073}a^{3}+\frac{891118034}{2971215073}a^{2}+\frac{9396}{2971215073}a+\frac{694359577}{2971215073}$, $\frac{1}{2971215073}a^{30}+\frac{647708023}{2971215073}a^{23}-\frac{27405}{2971215073}a^{22}+\frac{502385996}{2971215073}a^{21}-\frac{460404}{2971215073}a^{20}-\frac{180569404}{2971215073}a^{19}-\frac{3471300}{2971215073}a^{18}-\frac{631341483}{2971215073}a^{17}-\frac{15174540}{2971215073}a^{16}+\frac{957802915}{2971215073}a^{15}-\frac{41929650}{2971215073}a^{14}+\frac{492283050}{2971215073}a^{13}-\frac{75369840}{2971215073}a^{12}+\frac{582150800}{2971215073}a^{11}-\frac{87783696}{2971215073}a^{10}-\frac{1200157099}{2971215073}a^{9}-\frac{64127700}{2971215073}a^{8}+\frac{1305706370}{2971215073}a^{7}-\frac{27432405}{2971215073}a^{6}-\frac{482862484}{2971215073}a^{5}-\frac{6029100}{2971215073}a^{4}+\frac{518573452}{2971215073}a^{3}-\frac{516780}{2971215073}a^{2}+\frac{137204810}{2971215073}a-\frac{7308}{2971215073}$, $\frac{1}{2971215073}a^{31}-\frac{31465}{2971215073}a^{23}-\frac{186531191}{2971215073}a^{22}-\frac{553784}{2971215073}a^{21}+\frac{13837815}{2971215073}a^{20}-\frac{4405100}{2971215073}a^{19}+\frac{1295867793}{2971215073}a^{18}-\frac{20497200}{2971215073}a^{17}-\frac{287145316}{2971215073}a^{16}-\frac{60979170}{2971215073}a^{15}+\frac{787504972}{2971215073}a^{14}-\frac{119818720}{2971215073}a^{13}+\frac{1189699412}{2971215073}a^{12}-\frac{155764336}{2971215073}a^{11}+\frac{1021790391}{2971215073}a^{10}-\frac{130894400}{2971215073}a^{9}+\frac{175349539}{2971215073}a^{8}-\frac{67492425}{2971215073}a^{7}+\frac{412806790}{2971215073}a^{6}-\frac{19382440}{2971215073}a^{5}+\frac{286043286}{2971215073}a^{4}-\frac{2571140}{2971215073}a^{3}-\frac{1025083239}{2971215073}a^{2}-\frac{100688}{2971215073}a-\frac{1295416046}{2971215073}$, $\frac{1}{2971215073}a^{32}+\frac{1478965545}{2971215073}a^{23}+\frac{201376}{2971215073}a^{22}-\frac{305533206}{2971215073}a^{21}+\frac{3524080}{2971215073}a^{20}+\frac{1073372656}{2971215073}a^{19}+\frac{27329600}{2971215073}a^{18}-\frac{664003075}{2971215073}a^{17}+\frac{121958340}{2971215073}a^{16}+\frac{1002548294}{2971215073}a^{15}+\frac{342339200}{2971215073}a^{14}+\frac{701062139}{2971215073}a^{13}+\frac{623057344}{2971215073}a^{12}+\frac{274462797}{2971215073}a^{11}+\frac{733008640}{2971215073}a^{10}+\frac{608630145}{2971215073}a^{9}+\frac{539939400}{2971215073}a^{8}+\frac{78532139}{2971215073}a^{7}+\frac{232589280}{2971215073}a^{6}-\frac{815786622}{2971215073}a^{5}+\frac{51422800}{2971215073}a^{4}+\frac{862399518}{2971215073}a^{3}+\frac{4430272}{2971215073}a^{2}+\frac{1356428006}{2971215073}a+\frac{62930}{2971215073}$, $\frac{1}{2971215073}a^{33}+\frac{237336}{2971215073}a^{23}-\frac{146125410}{2971215073}a^{22}+\frac{4351160}{2971215073}a^{21}-\frac{224060559}{2971215073}a^{20}+\frac{35600400}{2971215073}a^{19}+\frac{518178786}{2971215073}a^{18}+\frac{169101900}{2971215073}a^{17}+\frac{993290926}{2971215073}a^{16}+\frac{511063520}{2971215073}a^{15}+\frac{208535882}{2971215073}a^{14}+\frac{1016747424}{2971215073}a^{13}+\frac{1260207390}{2971215073}a^{12}+\frac{1335122880}{2971215073}a^{11}-\frac{1244185757}{2971215073}a^{10}+\frac{1131301600}{2971215073}a^{9}-\frac{945677629}{2971215073}a^{8}+\frac{587406600}{2971215073}a^{7}-\frac{1108590004}{2971215073}a^{6}+\frac{169695240}{2971215073}a^{5}+\frac{375196640}{2971215073}a^{4}+\frac{22626032}{2971215073}a^{3}-\frac{658340291}{2971215073}a^{2}+\frac{890010}{2971215073}a+\frac{13283983}{2971215073}$, $\frac{1}{2971215073}a^{34}-\frac{993652788}{2971215073}a^{23}-\frac{1344904}{2971215073}a^{22}+\frac{1081315258}{2971215073}a^{21}-\frac{24208272}{2971215073}a^{20}-\frac{1284138409}{2971215073}a^{19}-\frac{191648820}{2971215073}a^{18}+\frac{1119335670}{2971215073}a^{17}-\frac{868807984}{2971215073}a^{16}-\frac{904154020}{2971215073}a^{15}+\frac{501971329}{2971215073}a^{14}-\frac{1336069048}{2971215073}a^{13}+\frac{1403012354}{2971215073}a^{12}+\frac{1251800732}{2971215073}a^{11}+\frac{557434530}{2971215073}a^{10}-\frac{563501396}{2971215073}a^{9}-\frac{1023149807}{2971215073}a^{8}+\frac{309201765}{2971215073}a^{7}+\frac{1240323625}{2971215073}a^{6}-\frac{142611128}{2971215073}a^{5}-\frac{384642544}{2971215073}a^{4}+\frac{1318992026}{2971215073}a^{3}-\frac{33286374}{2971215073}a^{2}+\frac{1318659800}{2971215073}a-\frac{474672}{2971215073}$, $\frac{1}{2971215073}a^{35}-\frac{1623160}{2971215073}a^{23}+\frac{1159261586}{2971215073}a^{22}-\frac{30608160}{2971215073}a^{21}-\frac{465701965}{2971215073}a^{20}-\frac{255647700}{2971215073}a^{19}-\frac{876898727}{2971215073}a^{18}-\frac{1233601600}{2971215073}a^{17}+\frac{151053500}{2971215073}a^{16}-\frac{803605823}{2971215073}a^{15}-\frac{1172357480}{2971215073}a^{14}+\frac{1327880739}{2971215073}a^{13}+\frac{427235014}{2971215073}a^{12}-\frac{1130468861}{2971215073}a^{11}-\frac{529354354}{2971215073}a^{10}+\frac{343360419}{2971215073}a^{9}+\frac{611762817}{2971215073}a^{8}+\frac{1465986746}{2971215073}a^{7}+\frac{114949682}{2971215073}a^{6}-\frac{1299826528}{2971215073}a^{5}+\frac{949724332}{2971215073}a^{4}-\frac{174083910}{2971215073}a^{3}-\frac{1184877305}{2971215073}a^{2}-\frac{6874560}{2971215073}a-\frac{983909497}{2971215073}$, $\frac{1}{2971215073}a^{36}+\frac{1013139370}{2971215073}a^{23}+\frac{8347680}{2971215073}a^{22}-\frac{855297860}{2971215073}a^{21}+\frac{153388620}{2971215073}a^{20}+\frac{1169572469}{2971215073}a^{19}+\frac{1233601600}{2971215073}a^{18}-\frac{1257407004}{2971215073}a^{17}-\frac{280198802}{2971215073}a^{16}-\frac{427113089}{2971215073}a^{15}+\frac{1399134235}{2971215073}a^{14}-\frac{805076480}{2971215073}a^{13}+\frac{420191510}{2971215073}a^{12}+\frac{1081481094}{2971215073}a^{11}+\frac{340615284}{2971215073}a^{10}+\frac{1344914754}{2971215073}a^{9}+\frac{117724743}{2971215073}a^{8}-\frac{633645185}{2971215073}a^{7}-\frac{186421540}{2971215073}a^{6}-\frac{1280617339}{2971215073}a^{5}-\frac{359956423}{2971215073}a^{4}-\frac{842341776}{2971215073}a^{3}+\frac{226860480}{2971215073}a^{2}-\frac{1373505392}{2971215073}a+\frac{3246320}{2971215073}$, $\frac{1}{2971215073}a^{37}+\frac{10295472}{2971215073}a^{23}-\frac{1400922156}{2971215073}a^{22}+\frac{198187836}{2971215073}a^{21}+\frac{1382947507}{2971215073}a^{20}-\frac{1289621313}{2971215073}a^{19}+\frac{831373483}{2971215073}a^{18}-\frac{697858563}{2971215073}a^{17}+\frac{1100079490}{2971215073}a^{16}-\frac{1346685993}{2971215073}a^{15}+\frac{1020157617}{2971215073}a^{14}+\frac{946112815}{2971215073}a^{13}+\frac{911010974}{2971215073}a^{12}+\frac{271078729}{2971215073}a^{11}+\frac{1105885460}{2971215073}a^{10}-\frac{534201620}{2971215073}a^{9}+\frac{219642524}{2971215073}a^{8}+\frac{1205151686}{2971215073}a^{7}-\frac{112327936}{2971215073}a^{6}+\frac{103901319}{2971215073}a^{5}-\frac{1228682991}{2971215073}a^{4}+\frac{1212443232}{2971215073}a^{3}+\frac{1295178978}{2971215073}a^{2}+\frac{48045536}{2971215073}a+\frac{944936333}{2971215073}$, $\frac{1}{2971215073}a^{38}+\frac{459720637}{2971215073}a^{23}-\frac{48903492}{2971215073}a^{22}-\frac{390494349}{2971215073}a^{21}-\frac{912865184}{2971215073}a^{20}+\frac{924245361}{2971215073}a^{19}+\frac{1480314435}{2971215073}a^{18}+\frac{1035206180}{2971215073}a^{17}+\frac{1190956332}{2971215073}a^{16}-\frac{1244957174}{2971215073}a^{15}+\frac{1258188802}{2971215073}a^{14}+\frac{577768250}{2971215073}a^{13}+\frac{962052063}{2971215073}a^{12}-\frac{452444026}{2971215073}a^{11}-\frac{941248917}{2971215073}a^{10}+\frac{917551655}{2971215073}a^{9}-\frac{1448740456}{2971215073}a^{8}-\frac{287825472}{2971215073}a^{7}+\frac{851783587}{2971215073}a^{6}+\frac{693627879}{2971215073}a^{5}+\frac{1372703718}{2971215073}a^{4}+\frac{905254095}{2971215073}a^{3}-\frac{1434502432}{2971215073}a^{2}-\frac{828505523}{2971215073}a-\frac{20590944}{2971215073}$, $\frac{1}{2971215073}a^{39}-\frac{61523748}{2971215073}a^{23}+\frac{461070655}{2971215073}a^{22}-\frac{1203131072}{2971215073}a^{21}+\frac{952032684}{2971215073}a^{20}-\frac{1422344445}{2971215073}a^{19}+\frac{495380095}{2971215073}a^{18}-\frac{498123919}{2971215073}a^{17}+\frac{32825008}{2971215073}a^{16}+\frac{1468249110}{2971215073}a^{15}-\frac{1198302150}{2971215073}a^{14}-\frac{528617267}{2971215073}a^{13}+\frac{296078540}{2971215073}a^{12}-\frac{1298545198}{2971215073}a^{11}+\frac{549372287}{2971215073}a^{10}-\frac{1003795266}{2971215073}a^{9}-\frac{175299706}{2971215073}a^{8}+\frac{1118750701}{2971215073}a^{7}+\frac{586242230}{2971215073}a^{6}+\frac{1461692756}{2971215073}a^{5}-\frac{603364652}{2971215073}a^{4}+\frac{1093293251}{2971215073}a^{3}+\frac{1309669428}{2971215073}a^{2}-\frac{310856832}{2971215073}a-\frac{919441274}{2971215073}$, $\frac{1}{2971215073}a^{40}-\frac{665861798}{2971215073}a^{23}+\frac{273438880}{2971215073}a^{22}-\frac{1197693151}{2971215073}a^{21}-\frac{774435314}{2971215073}a^{20}-\frac{203372744}{2971215073}a^{19}+\frac{910305778}{2971215073}a^{18}-\frac{681729594}{2971215073}a^{17}-\frac{349703851}{2971215073}a^{16}-\frac{2541686}{2971215073}a^{15}-\frac{117188835}{2971215073}a^{14}-\frac{875433808}{2971215073}a^{13}+\frac{275147922}{2971215073}a^{12}+\frac{1379307878}{2971215073}a^{11}+\frac{542068358}{2971215073}a^{10}-\frac{1111078990}{2971215073}a^{9}+\frac{348676641}{2971215073}a^{8}-\frac{569468355}{2971215073}a^{7}+\frac{922164622}{2971215073}a^{6}+\frac{992208535}{2971215073}a^{5}-\frac{295697809}{2971215073}a^{4}-\frac{1416072847}{2971215073}a^{3}-\frac{365082339}{2971215073}a^{2}-\frac{97952036}{2971215073}a+\frac{123047496}{2971215073}$, $\frac{1}{2971215073}a^{41}+\frac{350343565}{2971215073}a^{23}-\frac{73085364}{2971215073}a^{22}+\frac{994372441}{2971215073}a^{21}+\frac{1205756264}{2971215073}a^{20}+\frac{771092890}{2971215073}a^{19}+\frac{1215078546}{2971215073}a^{18}-\frac{548974298}{2971215073}a^{17}-\frac{175288233}{2971215073}a^{16}+\frac{1202569352}{2971215073}a^{15}+\frac{1033849973}{2971215073}a^{14}+\frac{1373773643}{2971215073}a^{13}+\frac{1460522043}{2971215073}a^{12}-\frac{1273227684}{2971215073}a^{11}-\frac{1095897271}{2971215073}a^{10}-\frac{691399632}{2971215073}a^{9}+\frac{416136237}{2971215073}a^{8}-\frac{890367171}{2971215073}a^{7}-\frac{117569116}{2971215073}a^{6}+\frac{90529951}{2971215073}a^{5}+\frac{256184489}{2971215073}a^{4}-\frac{77107678}{2971215073}a^{3}+\frac{707264540}{2971215073}a^{2}-\frac{1079359822}{2971215073}a+\frac{1331723596}{2971215073}$, $\frac{1}{2971215073}a^{42}-\frac{341065032}{2971215073}a^{23}-\frac{1471442973}{2971215073}a^{22}+\frac{984654046}{2971215073}a^{21}-\frac{1350248373}{2971215073}a^{20}-\frac{995943634}{2971215073}a^{19}-\frac{1223695031}{2971215073}a^{18}-\frac{893254367}{2971215073}a^{17}-\frac{412592553}{2971215073}a^{16}+\frac{497223596}{2971215073}a^{15}-\frac{1299217714}{2971215073}a^{14}+\frac{208393830}{2971215073}a^{13}-\frac{350477}{2971215073}a^{12}+\frac{1358517040}{2971215073}a^{11}+\frac{1070086102}{2971215073}a^{10}-\frac{197550764}{2971215073}a^{9}+\frac{1183831725}{2971215073}a^{8}+\frac{108461698}{2971215073}a^{7}-\frac{633709657}{2971215073}a^{6}+\frac{1321933118}{2971215073}a^{5}-\frac{1081220472}{2971215073}a^{4}-\frac{1185769183}{2971215073}a^{3}-\frac{1018176941}{2971215073}a^{2}+\frac{1110621378}{2971215073}a-\frac{700687130}{2971215073}$, $\frac{1}{2971215073}a^{43}+\frac{1053880290}{2971215073}a^{23}+\frac{256569595}{2971215073}a^{22}+\frac{279100289}{2971215073}a^{21}-\frac{1212792687}{2971215073}a^{20}+\frac{213716224}{2971215073}a^{19}+\frac{534171571}{2971215073}a^{18}+\frac{352613918}{2971215073}a^{17}-\frac{1322349120}{2971215073}a^{16}+\frac{1283036231}{2971215073}a^{15}+\frac{302970768}{2971215073}a^{14}+\frac{82478582}{2971215073}a^{13}-\frac{793048362}{2971215073}a^{12}-\frac{1250117162}{2971215073}a^{11}-\frac{1185942268}{2971215073}a^{10}+\frac{1451816439}{2971215073}a^{9}+\frac{156302690}{2971215073}a^{8}+\frac{121324186}{2971215073}a^{7}-\frac{947157786}{2971215073}a^{6}+\frac{160862500}{2971215073}a^{5}-\frac{1247543652}{2971215073}a^{4}-\frac{827087253}{2971215073}a^{3}-\frac{286670255}{2971215073}a^{2}+\frac{928661532}{2971215073}a+\frac{682130064}{2971215073}$, $\frac{1}{2971215073}a^{44}+\frac{1128906218}{2971215073}a^{23}-\frac{1244306087}{2971215073}a^{22}+\frac{1023659204}{2971215073}a^{21}-\frac{925975359}{2971215073}a^{20}-\frac{871030103}{2971215073}a^{19}-\frac{60502535}{2971215073}a^{18}+\frac{1364375601}{2971215073}a^{17}+\frac{668510697}{2971215073}a^{16}-\frac{1027964710}{2971215073}a^{15}+\frac{719309392}{2971215073}a^{14}+\frac{63055620}{2971215073}a^{13}+\frac{373285698}{2971215073}a^{12}-\frac{1274825624}{2971215073}a^{11}-\frac{193044927}{2971215073}a^{10}+\frac{705695481}{2971215073}a^{9}-\frac{1128069433}{2971215073}a^{8}-\frac{1211765126}{2971215073}a^{7}-\frac{1061692500}{2971215073}a^{6}+\frac{645063950}{2971215073}a^{5}+\frac{184314564}{2971215073}a^{4}+\frac{1375830179}{2971215073}a^{3}+\frac{701868495}{2971215073}a^{2}-\frac{52633118}{2971215073}a+\frac{863454493}{2971215073}$, $\frac{1}{2971215073}a^{45}+\frac{1236950691}{2971215073}a^{23}+\frac{670845629}{2971215073}a^{22}-\frac{310155852}{2971215073}a^{21}-\frac{118750031}{2971215073}a^{20}+\frac{155262389}{2971215073}a^{19}-\frac{181978638}{2971215073}a^{18}+\frac{115641720}{2971215073}a^{17}-\frac{1074619905}{2971215073}a^{16}-\frac{1415759319}{2971215073}a^{15}+\frac{1039848049}{2971215073}a^{14}-\frac{646921197}{2971215073}a^{13}+\frac{316228005}{2971215073}a^{12}-\frac{529919854}{2971215073}a^{11}+\frac{1172215609}{2971215073}a^{10}-\frac{556952732}{2971215073}a^{9}-\frac{883743161}{2971215073}a^{8}-\frac{1313265494}{2971215073}a^{7}-\frac{1199677728}{2971215073}a^{6}-\frac{889948125}{2971215073}a^{5}+\frac{1404987687}{2971215073}a^{4}-\frac{606177716}{2971215073}a^{3}+\frac{801700505}{2971215073}a^{2}+\frac{1479274000}{2971215073}a+\frac{713402637}{2971215073}$ 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, a^23, 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631341483/2971215073*a^17 - 15174540/2971215073*a^16 + 957802915/2971215073*a^15 - 41929650/2971215073*a^14 + 492283050/2971215073*a^13 - 75369840/2971215073*a^12 + 582150800/2971215073*a^11 - 87783696/2971215073*a^10 - 1200157099/2971215073*a^9 - 64127700/2971215073*a^8 + 1305706370/2971215073*a^7 - 27432405/2971215073*a^6 - 482862484/2971215073*a^5 - 6029100/2971215073*a^4 + 518573452/2971215073*a^3 - 516780/2971215073*a^2 + 137204810/2971215073*a - 7308/2971215073, 1/2971215073*a^31 - 31465/2971215073*a^23 - 186531191/2971215073*a^22 - 553784/2971215073*a^21 + 13837815/2971215073*a^20 - 4405100/2971215073*a^19 + 1295867793/2971215073*a^18 - 20497200/2971215073*a^17 - 287145316/2971215073*a^16 - 60979170/2971215073*a^15 + 787504972/2971215073*a^14 - 119818720/2971215073*a^13 + 1189699412/2971215073*a^12 - 155764336/2971215073*a^11 + 1021790391/2971215073*a^10 - 130894400/2971215073*a^9 + 175349539/2971215073*a^8 - 67492425/2971215073*a^7 + 412806790/2971215073*a^6 - 19382440/2971215073*a^5 + 286043286/2971215073*a^4 - 2571140/2971215073*a^3 - 1025083239/2971215073*a^2 - 100688/2971215073*a - 1295416046/2971215073, 1/2971215073*a^32 + 1478965545/2971215073*a^23 + 201376/2971215073*a^22 - 305533206/2971215073*a^21 + 3524080/2971215073*a^20 + 1073372656/2971215073*a^19 + 27329600/2971215073*a^18 - 664003075/2971215073*a^17 + 121958340/2971215073*a^16 + 1002548294/2971215073*a^15 + 342339200/2971215073*a^14 + 701062139/2971215073*a^13 + 623057344/2971215073*a^12 + 274462797/2971215073*a^11 + 733008640/2971215073*a^10 + 608630145/2971215073*a^9 + 539939400/2971215073*a^8 + 78532139/2971215073*a^7 + 232589280/2971215073*a^6 - 815786622/2971215073*a^5 + 51422800/2971215073*a^4 + 862399518/2971215073*a^3 + 4430272/2971215073*a^2 + 1356428006/2971215073*a + 62930/2971215073, 1/2971215073*a^33 + 237336/2971215073*a^23 - 146125410/2971215073*a^22 + 4351160/2971215073*a^21 - 224060559/2971215073*a^20 + 35600400/2971215073*a^19 + 518178786/2971215073*a^18 + 169101900/2971215073*a^17 + 993290926/2971215073*a^16 + 511063520/2971215073*a^15 + 208535882/2971215073*a^14 + 1016747424/2971215073*a^13 + 1260207390/2971215073*a^12 + 1335122880/2971215073*a^11 - 1244185757/2971215073*a^10 + 1131301600/2971215073*a^9 - 945677629/2971215073*a^8 + 587406600/2971215073*a^7 - 1108590004/2971215073*a^6 + 169695240/2971215073*a^5 + 375196640/2971215073*a^4 + 22626032/2971215073*a^3 - 658340291/2971215073*a^2 + 890010/2971215073*a + 13283983/2971215073, 1/2971215073*a^34 - 993652788/2971215073*a^23 - 1344904/2971215073*a^22 + 1081315258/2971215073*a^21 - 24208272/2971215073*a^20 - 1284138409/2971215073*a^19 - 191648820/2971215073*a^18 + 1119335670/2971215073*a^17 - 868807984/2971215073*a^16 - 904154020/2971215073*a^15 + 501971329/2971215073*a^14 - 1336069048/2971215073*a^13 + 1403012354/2971215073*a^12 + 1251800732/2971215073*a^11 + 557434530/2971215073*a^10 - 563501396/2971215073*a^9 - 1023149807/2971215073*a^8 + 309201765/2971215073*a^7 + 1240323625/2971215073*a^6 - 142611128/2971215073*a^5 - 384642544/2971215073*a^4 + 1318992026/2971215073*a^3 - 33286374/2971215073*a^2 + 1318659800/2971215073*a - 474672/2971215073, 1/2971215073*a^35 - 1623160/2971215073*a^23 + 1159261586/2971215073*a^22 - 30608160/2971215073*a^21 - 465701965/2971215073*a^20 - 255647700/2971215073*a^19 - 876898727/2971215073*a^18 - 1233601600/2971215073*a^17 + 151053500/2971215073*a^16 - 803605823/2971215073*a^15 - 1172357480/2971215073*a^14 + 1327880739/2971215073*a^13 + 427235014/2971215073*a^12 - 1130468861/2971215073*a^11 - 529354354/2971215073*a^10 + 343360419/2971215073*a^9 + 611762817/2971215073*a^8 + 1465986746/2971215073*a^7 + 114949682/2971215073*a^6 - 1299826528/2971215073*a^5 + 949724332/2971215073*a^4 - 174083910/2971215073*a^3 - 1184877305/2971215073*a^2 - 6874560/2971215073*a - 983909497/2971215073, 1/2971215073*a^36 + 1013139370/2971215073*a^23 + 8347680/2971215073*a^22 - 855297860/2971215073*a^21 + 153388620/2971215073*a^20 + 1169572469/2971215073*a^19 + 1233601600/2971215073*a^18 - 1257407004/2971215073*a^17 - 280198802/2971215073*a^16 - 427113089/2971215073*a^15 + 1399134235/2971215073*a^14 - 805076480/2971215073*a^13 + 420191510/2971215073*a^12 + 1081481094/2971215073*a^11 + 340615284/2971215073*a^10 + 1344914754/2971215073*a^9 + 117724743/2971215073*a^8 - 633645185/2971215073*a^7 - 186421540/2971215073*a^6 - 1280617339/2971215073*a^5 - 359956423/2971215073*a^4 - 842341776/2971215073*a^3 + 226860480/2971215073*a^2 - 1373505392/2971215073*a + 3246320/2971215073, 1/2971215073*a^37 + 10295472/2971215073*a^23 - 1400922156/2971215073*a^22 + 198187836/2971215073*a^21 + 1382947507/2971215073*a^20 - 1289621313/2971215073*a^19 + 831373483/2971215073*a^18 - 697858563/2971215073*a^17 + 1100079490/2971215073*a^16 - 1346685993/2971215073*a^15 + 1020157617/2971215073*a^14 + 946112815/2971215073*a^13 + 911010974/2971215073*a^12 + 271078729/2971215073*a^11 + 1105885460/2971215073*a^10 - 534201620/2971215073*a^9 + 219642524/2971215073*a^8 + 1205151686/2971215073*a^7 - 112327936/2971215073*a^6 + 103901319/2971215073*a^5 - 1228682991/2971215073*a^4 + 1212443232/2971215073*a^3 + 1295178978/2971215073*a^2 + 48045536/2971215073*a + 944936333/2971215073, 1/2971215073*a^38 + 459720637/2971215073*a^23 - 48903492/2971215073*a^22 - 390494349/2971215073*a^21 - 912865184/2971215073*a^20 + 924245361/2971215073*a^19 + 1480314435/2971215073*a^18 + 1035206180/2971215073*a^17 + 1190956332/2971215073*a^16 - 1244957174/2971215073*a^15 + 1258188802/2971215073*a^14 + 577768250/2971215073*a^13 + 962052063/2971215073*a^12 - 452444026/2971215073*a^11 - 941248917/2971215073*a^10 + 917551655/2971215073*a^9 - 1448740456/2971215073*a^8 - 287825472/2971215073*a^7 + 851783587/2971215073*a^6 + 693627879/2971215073*a^5 + 1372703718/2971215073*a^4 + 905254095/2971215073*a^3 - 1434502432/2971215073*a^2 - 828505523/2971215073*a - 20590944/2971215073, 1/2971215073*a^39 - 61523748/2971215073*a^23 + 461070655/2971215073*a^22 - 1203131072/2971215073*a^21 + 952032684/2971215073*a^20 - 1422344445/2971215073*a^19 + 495380095/2971215073*a^18 - 498123919/2971215073*a^17 + 32825008/2971215073*a^16 + 1468249110/2971215073*a^15 - 1198302150/2971215073*a^14 - 528617267/2971215073*a^13 + 296078540/2971215073*a^12 - 1298545198/2971215073*a^11 + 549372287/2971215073*a^10 - 1003795266/2971215073*a^9 - 175299706/2971215073*a^8 + 1118750701/2971215073*a^7 + 586242230/2971215073*a^6 + 1461692756/2971215073*a^5 - 603364652/2971215073*a^4 + 1093293251/2971215073*a^3 + 1309669428/2971215073*a^2 - 310856832/2971215073*a - 919441274/2971215073, 1/2971215073*a^40 - 665861798/2971215073*a^23 + 273438880/2971215073*a^22 - 1197693151/2971215073*a^21 - 774435314/2971215073*a^20 - 203372744/2971215073*a^19 + 910305778/2971215073*a^18 - 681729594/2971215073*a^17 - 349703851/2971215073*a^16 - 2541686/2971215073*a^15 - 117188835/2971215073*a^14 - 875433808/2971215073*a^13 + 275147922/2971215073*a^12 + 1379307878/2971215073*a^11 + 542068358/2971215073*a^10 - 1111078990/2971215073*a^9 + 348676641/2971215073*a^8 - 569468355/2971215073*a^7 + 922164622/2971215073*a^6 + 992208535/2971215073*a^5 - 295697809/2971215073*a^4 - 1416072847/2971215073*a^3 - 365082339/2971215073*a^2 - 97952036/2971215073*a + 123047496/2971215073, 1/2971215073*a^41 + 350343565/2971215073*a^23 - 73085364/2971215073*a^22 + 994372441/2971215073*a^21 + 1205756264/2971215073*a^20 + 771092890/2971215073*a^19 + 1215078546/2971215073*a^18 - 548974298/2971215073*a^17 - 175288233/2971215073*a^16 + 1202569352/2971215073*a^15 + 1033849973/2971215073*a^14 + 1373773643/2971215073*a^13 + 1460522043/2971215073*a^12 - 1273227684/2971215073*a^11 - 1095897271/2971215073*a^10 - 691399632/2971215073*a^9 + 416136237/2971215073*a^8 - 890367171/2971215073*a^7 - 117569116/2971215073*a^6 + 90529951/2971215073*a^5 + 256184489/2971215073*a^4 - 77107678/2971215073*a^3 + 707264540/2971215073*a^2 - 1079359822/2971215073*a + 1331723596/2971215073, 1/2971215073*a^42 - 341065032/2971215073*a^23 - 1471442973/2971215073*a^22 + 984654046/2971215073*a^21 - 1350248373/2971215073*a^20 - 995943634/2971215073*a^19 - 1223695031/2971215073*a^18 - 893254367/2971215073*a^17 - 412592553/2971215073*a^16 + 497223596/2971215073*a^15 - 1299217714/2971215073*a^14 + 208393830/2971215073*a^13 - 350477/2971215073*a^12 + 1358517040/2971215073*a^11 + 1070086102/2971215073*a^10 - 197550764/2971215073*a^9 + 1183831725/2971215073*a^8 + 108461698/2971215073*a^7 - 633709657/2971215073*a^6 + 1321933118/2971215073*a^5 - 1081220472/2971215073*a^4 - 1185769183/2971215073*a^3 - 1018176941/2971215073*a^2 + 1110621378/2971215073*a - 700687130/2971215073, 1/2971215073*a^43 + 1053880290/2971215073*a^23 + 256569595/2971215073*a^22 + 279100289/2971215073*a^21 - 1212792687/2971215073*a^20 + 213716224/2971215073*a^19 + 534171571/2971215073*a^18 + 352613918/2971215073*a^17 - 1322349120/2971215073*a^16 + 1283036231/2971215073*a^15 + 302970768/2971215073*a^14 + 82478582/2971215073*a^13 - 793048362/2971215073*a^12 - 1250117162/2971215073*a^11 - 1185942268/2971215073*a^10 + 1451816439/2971215073*a^9 + 156302690/2971215073*a^8 + 121324186/2971215073*a^7 - 947157786/2971215073*a^6 + 160862500/2971215073*a^5 - 1247543652/2971215073*a^4 - 827087253/2971215073*a^3 - 286670255/2971215073*a^2 + 928661532/2971215073*a + 682130064/2971215073, 1/2971215073*a^44 + 1128906218/2971215073*a^23 - 1244306087/2971215073*a^22 + 1023659204/2971215073*a^21 - 925975359/2971215073*a^20 - 871030103/2971215073*a^19 - 60502535/2971215073*a^18 + 1364375601/2971215073*a^17 + 668510697/2971215073*a^16 - 1027964710/2971215073*a^15 + 719309392/2971215073*a^14 + 63055620/2971215073*a^13 + 373285698/2971215073*a^12 - 1274825624/2971215073*a^11 - 193044927/2971215073*a^10 + 705695481/2971215073*a^9 - 1128069433/2971215073*a^8 - 1211765126/2971215073*a^7 - 1061692500/2971215073*a^6 + 645063950/2971215073*a^5 + 184314564/2971215073*a^4 + 1375830179/2971215073*a^3 + 701868495/2971215073*a^2 - 52633118/2971215073*a + 863454493/2971215073, 1/2971215073*a^45 + 1236950691/2971215073*a^23 + 670845629/2971215073*a^22 - 310155852/2971215073*a^21 - 118750031/2971215073*a^20 + 155262389/2971215073*a^19 - 181978638/2971215073*a^18 + 115641720/2971215073*a^17 - 1074619905/2971215073*a^16 - 1415759319/2971215073*a^15 + 1039848049/2971215073*a^14 - 646921197/2971215073*a^13 + 316228005/2971215073*a^12 - 529919854/2971215073*a^11 + 1172215609/2971215073*a^10 - 556952732/2971215073*a^9 - 883743161/2971215073*a^8 - 1313265494/2971215073*a^7 - 1199677728/2971215073*a^6 - 889948125/2971215073*a^5 + 1404987687/2971215073*a^4 - 606177716/2971215073*a^3 + 801700505/2971215073*a^2 + 1479274000/2971215073*a + 713402637/2971215073

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:	$22$	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^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879)

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^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879, 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^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879);

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^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879);

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

$C_{46}$ (as 46T1):

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 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$

Intermediate fields

$\Q(\sqrt{-235}) $, $\Q(\zeta_{47})^+$

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	$46$	$46$	R	$46$	$46$	$23^{2}$	$46$	$46$	$23^{2}$	$46$	$46$	$46$	$46$	$23^{2}$	R	$46$	$23^{2}$

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
$5$ 5		Deg $46$	$2$	$23$	$23$
$47$ 47		Deg $46$	$46$	$1$	$45$

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Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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