LMFDB - Number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)

gp: K = bnfinit(y^44 - y^43 - 23*y^42 + 24*y^41 + 298*y^40 - 322*y^39 - 2644*y^38 + 2966*y^37 + 17733*y^36 - 20699*y^35 - 93680*y^34 + 114379*y^33 + 400546*y^32 - 514925*y^31 - 1402310*y^30 + 1917235*y^29 + 4042594*y^28 - 5959829*y^27 - 9550726*y^26 + 15510555*y^25 + 18239299*y^24 - 33749854*y^23 - 27151731*y^22 + 60965664*y^21 + 28718628*y^20 - 92631926*y^19 - 11478724*y^18 + 118848820*y^17 - 33434940*y^16 - 1406311*y^15 - 22927136*y^14 - 576983889*y^13 + 635212253*y^12 + 643308528*y^11 - 1294716116*y^10 + 1724346364*y^9 - 424183664*y^8 - 3407721010*y^7 + 3830694322*y^6 + 209294181*y^5 - 4039808114*y^4 + 4041269764*y^3 - 1473288*y^2 - 4104644424*y + 4106118241, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)

$ x^{44} - x^{43} - 23 x^{42} + 24 x^{41} + 298 x^{40} - 322 x^{39} - 2644 x^{38} + 2966 x^{37} + \cdots + 4106118241 $ x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241

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:	$116\!\cdots\!625$ 116567320065927752512435466812933331534234135648947894549180382317450046539306640625 $\medspace = 3^{22}\cdot 5^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$77.25$	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:	$3^{1/2}5^{1/2}23^{21/22}\approx 77.24613267922518$
Ramified primes:	$3$, $5$, $23$ 3, 5, 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:	$345=3\cdot 5\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(131,·)$, $\chi_{345}(134,·)$, $\chi_{345}(266,·)$, $\chi_{345}(109,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(146,·)$, $\chi_{345}(19,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(26,·)$, $\chi_{345}(31,·)$, $\chi_{345}(34,·)$, $\chi_{345}(41,·)$, $\chi_{345}(71,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(304,·)$, $\chi_{345}(311,·)$, $\chi_{345}(116,·)$, $\chi_{345}(314,·)$, $\chi_{345}(319,·)$, $\chi_{345}(194,·)$, $\chi_{345}(196,·)$, $\chi_{345}(326,·)$, $\chi_{345}(199,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(79,·)$, $\chi_{345}(211,·)$, $\chi_{345}(214,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(101,·)$, $\chi_{345}(224,·)$, $\chi_{345}(229,·)$, $\chi_{345}(236,·)$, $\chi_{345}(274,·)$, $\chi_{345}(244,·)$, $\chi_{345}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{28657}a^{24}+\frac{10946}{28657}a^{23}-\frac{13}{28657}a^{22}+\frac{5168}{28657}a^{21}+\frac{101}{28657}a^{20}-\frac{754}{28657}a^{19}-\frac{501}{28657}a^{18}-\frac{7930}{28657}a^{17}+\frac{1797}{28657}a^{16}+\frac{11857}{28657}a^{15}-\frac{4598}{28657}a^{14}-\frac{3796}{28657}a^{13}+\frac{8635}{28657}a^{12}+\frac{9498}{28657}a^{11}-\frac{11284}{28657}a^{10}+\frac{14025}{28657}a^{9}+\frac{10152}{28657}a^{8}+\frac{12302}{28657}a^{7}-\frac{5510}{28657}a^{6}-\frac{6477}{28657}a^{5}+\frac{1691}{28657}a^{4}+\frac{6477}{28657}a^{3}-\frac{193}{28657}a^{2}+\frac{11556}{28657}a+\frac{4}{28657}$, $\frac{1}{28657}a^{25}-\frac{12}{28657}a^{23}+\frac{4181}{28657}a^{22}+\frac{91}{28657}a^{21}+\frac{11323}{28657}a^{20}-\frac{433}{28657}a^{19}+\frac{2529}{28657}a^{18}+\frac{1524}{28657}a^{17}+\frac{597}{28657}a^{16}-\frac{3767}{28657}a^{15}+\frac{4220}{28657}a^{14}+\frac{7001}{28657}a^{13}+\frac{1574}{28657}a^{12}-\frac{8796}{28657}a^{11}-\frac{11638}{28657}a^{10}+\frac{8051}{28657}a^{9}-\frac{8301}{28657}a^{8}-\frac{3959}{28657}a^{7}+\frac{11655}{28657}a^{6}+\frac{1515}{28657}a^{5}+\frac{9213}{28657}a^{4}-\frac{17}{28657}a^{3}+\frac{3516}{28657}a^{2}+\frac{26}{28657}a+\frac{13530}{28657}$, $\frac{1}{28657}a^{26}-\frac{7752}{28657}a^{23}-\frac{65}{28657}a^{22}-\frac{12632}{28657}a^{21}+\frac{779}{28657}a^{20}-\frac{6519}{28657}a^{19}-\frac{4488}{28657}a^{18}-\frac{8592}{28657}a^{17}-\frac{10860}{28657}a^{16}+\frac{3219}{28657}a^{15}+\frac{9139}{28657}a^{14}+\frac{13336}{28657}a^{13}+\frac{8853}{28657}a^{12}-\frac{12290}{28657}a^{11}-\frac{12729}{28657}a^{10}-\frac{11943}{28657}a^{9}+\frac{3237}{28657}a^{8}-\frac{12663}{28657}a^{7}-\frac{7291}{28657}a^{6}-\frac{11197}{28657}a^{5}-\frac{8382}{28657}a^{4}-\frac{4731}{28657}a^{3}-\frac{2290}{28657}a^{2}+\frac{8917}{28657}a+\frac{48}{28657}$, $\frac{1}{28657}a^{27}-\frac{50}{28657}a^{23}+\frac{1220}{28657}a^{22}+\frac{629}{28657}a^{21}+\frac{2694}{28657}a^{20}-\frac{3468}{28657}a^{19}+\frac{5008}{28657}a^{18}+\frac{13702}{28657}a^{17}+\frac{6261}{28657}a^{16}-\frac{7053}{28657}a^{15}-\frac{9709}{28657}a^{14}+\frac{13000}{28657}a^{13}+\frac{12135}{28657}a^{12}-\frac{4066}{28657}a^{11}+\frac{4310}{28657}a^{10}+\frac{379}{28657}a^{9}-\frac{6481}{28657}a^{8}-\frac{12683}{28657}a^{7}+\frac{2870}{28657}a^{6}-\frac{11022}{28657}a^{5}+\frac{7652}{28657}a^{4}+\frac{350}{28657}a^{3}+\frac{2945}{28657}a^{2}+\frac{378}{28657}a+\frac{2351}{28657}$, $\frac{1}{28657}a^{28}+\frac{4037}{28657}a^{23}-\frac{21}{28657}a^{22}+\frac{3181}{28657}a^{21}+\frac{1582}{28657}a^{20}-\frac{4035}{28657}a^{19}-\frac{11348}{28657}a^{18}+\frac{10959}{28657}a^{17}-\frac{3174}{28657}a^{16}+\frac{10001}{28657}a^{15}+\frac{12356}{28657}a^{14}-\frac{5723}{28657}a^{13}-\frac{2171}{28657}a^{12}-\frac{7959}{28657}a^{11}+\frac{9319}{28657}a^{10}+\frac{7001}{28657}a^{9}+\frac{7748}{28657}a^{8}-\frac{12484}{28657}a^{7}+\frac{48}{28657}a^{6}-\frac{971}{28657}a^{5}-\frac{1071}{28657}a^{4}+\frac{11568}{28657}a^{3}-\frac{9272}{28657}a^{2}+\frac{7011}{28657}a+\frac{200}{28657}$, $\frac{1}{28657}a^{29}+\frac{71}{28657}a^{23}-\frac{1652}{28657}a^{22}+\frac{662}{28657}a^{21}-\frac{10574}{28657}a^{20}-\frac{5092}{28657}a^{19}-\frac{1151}{28657}a^{18}+\frac{367}{28657}a^{17}+\frac{5733}{28657}a^{16}+\frac{2837}{28657}a^{15}-\frac{13333}{28657}a^{14}-\frac{9214}{28657}a^{13}+\frac{8115}{28657}a^{12}+\frac{8959}{28657}a^{11}-\frac{4121}{28657}a^{10}-\frac{13602}{28657}a^{9}+\frac{12059}{28657}a^{8}-\frac{545}{28657}a^{7}+\frac{5067}{28657}a^{6}+\frac{11394}{28657}a^{5}+\frac{5367}{28657}a^{4}+\frac{6920}{28657}a^{3}+\frac{12413}{28657}a^{2}+\frac{2224}{28657}a+\frac{12509}{28657}$, $\frac{1}{28657}a^{30}-\frac{5079}{28657}a^{23}+\frac{1585}{28657}a^{22}-\frac{4961}{28657}a^{21}-\frac{12263}{28657}a^{20}-\frac{4931}{28657}a^{19}+\frac{7281}{28657}a^{18}-\frac{4377}{28657}a^{17}-\frac{10122}{28657}a^{16}+\frac{4530}{28657}a^{15}+\frac{2017}{28657}a^{14}-\frac{8939}{28657}a^{13}-\frac{2329}{28657}a^{12}+\frac{9289}{28657}a^{11}+\frac{13823}{28657}a^{10}-\frac{9378}{28657}a^{9}-\frac{4912}{28657}a^{8}-\frac{8665}{28657}a^{7}+\frac{1406}{28657}a^{6}+\frac{6722}{28657}a^{5}+\frac{1487}{28657}a^{4}+\frac{11058}{28657}a^{3}-\frac{12730}{28657}a^{2}-\frac{5571}{28657}a-\frac{284}{28657}$, $\frac{1}{28657}a^{31}+\frac{1739}{28657}a^{23}-\frac{13674}{28657}a^{22}-\frac{13803}{28657}a^{21}-\frac{7778}{28657}a^{20}-\frac{10904}{28657}a^{19}+\frac{1517}{28657}a^{18}+\frac{5150}{28657}a^{17}-\frac{10090}{28657}a^{16}-\frac{13294}{28657}a^{15}-\frac{6726}{28657}a^{14}+\frac{3948}{28657}a^{13}-\frac{7413}{28657}a^{12}-\frac{4223}{28657}a^{11}-\frac{6814}{28657}a^{10}-\frac{13239}{28657}a^{9}-\frac{600}{28657}a^{8}+\frac{11004}{28657}a^{7}-\frac{9336}{28657}a^{6}+\frac{3040}{28657}a^{5}+\frac{2547}{28657}a^{4}-\frac{14283}{28657}a^{3}-\frac{11480}{28657}a^{2}+\frac{3104}{28657}a-\frac{8341}{28657}$, $\frac{1}{28657}a^{32}+\frac{8137}{28657}a^{23}+\frac{8804}{28657}a^{22}+\frac{3368}{28657}a^{21}+\frac{14056}{28657}a^{20}-\frac{5499}{28657}a^{19}-\frac{11978}{28657}a^{18}-\frac{3837}{28657}a^{17}+\frac{13993}{28657}a^{16}+\frac{6991}{28657}a^{15}+\frac{4567}{28657}a^{14}+\frac{2721}{28657}a^{13}-\frac{4220}{28657}a^{12}+\frac{11253}{28657}a^{11}+\frac{8249}{28657}a^{10}-\frac{2968}{28657}a^{9}+\frac{9388}{28657}a^{8}+\frac{4265}{28657}a^{7}+\frac{13492}{28657}a^{6}+\frac{3849}{28657}a^{5}-\frac{3261}{28657}a^{4}-\frac{12782}{28657}a^{3}-\frac{5153}{28657}a^{2}+\frac{12989}{28657}a-\frac{6956}{28657}$, $\frac{1}{28657}a^{33}+\frac{7158}{28657}a^{23}-\frac{5479}{28657}a^{22}+\frac{1859}{28657}a^{21}+\frac{3717}{28657}a^{20}-\frac{9278}{28657}a^{19}+\frac{3506}{28657}a^{18}+\frac{4839}{28657}a^{17}-\frac{128}{28657}a^{16}+\frac{12277}{28657}a^{15}-\frac{9395}{28657}a^{14}-\frac{8414}{28657}a^{13}-\frac{13435}{28657}a^{12}+\frac{10952}{28657}a^{11}-\frac{2088}{28657}a^{10}+\frac{137}{28657}a^{9}-\frac{13085}{28657}a^{8}+\frac{11019}{28657}a^{7}-\frac{9486}{28657}a^{6}-\frac{135}{28657}a^{5}+\frac{11568}{28657}a^{4}-\frac{8279}{28657}a^{3}+\frac{7295}{28657}a^{2}+\frac{14146}{28657}a-\frac{3891}{28657}$, $\frac{1}{28657}a^{34}-\frac{8709}{28657}a^{23}+\frac{8942}{28657}a^{22}+\frac{7360}{28657}a^{21}+\frac{12846}{28657}a^{20}+\frac{13122}{28657}a^{19}+\frac{8872}{28657}a^{18}-\frac{6705}{28657}a^{17}-\frac{12313}{28657}a^{16}+\frac{233}{28657}a^{15}+\frac{5834}{28657}a^{14}-\frac{8503}{28657}a^{13}-\frac{13886}{28657}a^{12}+\frac{14289}{28657}a^{11}-\frac{13074}{28657}a^{10}+\frac{10093}{28657}a^{9}-\frac{11502}{28657}a^{8}-\frac{4241}{28657}a^{7}+\frac{8413}{28657}a^{6}+\frac{6908}{28657}a^{5}+\frac{9454}{28657}a^{4}+\frac{11955}{28657}a^{3}-\frac{8553}{28657}a^{2}+\frac{11020}{28657}a+\frac{25}{28657}$, $\frac{1}{28657}a^{35}-\frac{4183}{28657}a^{23}+\frac{8771}{28657}a^{22}+\frac{811}{28657}a^{21}+\frac{4364}{28657}a^{20}+\frac{4739}{28657}a^{19}-\frac{14050}{28657}a^{18}-\frac{11313}{28657}a^{17}+\frac{3584}{28657}a^{16}-\frac{11381}{28657}a^{15}+\frac{10001}{28657}a^{14}-\frac{3072}{28657}a^{13}-\frac{8121}{28657}a^{12}+\frac{906}{28657}a^{11}+\frac{2590}{28657}a^{10}-\frac{3911}{28657}a^{9}+\frac{2682}{28657}a^{8}-\frac{1992}{28657}a^{7}-\frac{7864}{28657}a^{6}-\frac{1763}{28657}a^{5}+\frac{9176}{28657}a^{4}+\frac{2664}{28657}a^{3}-\frac{7711}{28657}a^{2}-\frac{2155}{28657}a+\frac{6179}{28657}$, $\frac{1}{28657}a^{36}+\frac{2003}{28657}a^{23}+\frac{3746}{28657}a^{22}-\frac{13927}{28657}a^{21}-\frac{2633}{28657}a^{20}+\frac{12895}{28657}a^{19}+\frac{13622}{28657}a^{18}-\frac{11457}{28657}a^{17}-\frac{2664}{28657}a^{16}+\frac{2565}{28657}a^{15}-\frac{7659}{28657}a^{14}-\frac{10811}{28657}a^{13}+\frac{13291}{28657}a^{12}+\frac{14122}{28657}a^{11}-\frac{6804}{28657}a^{10}+\frac{8378}{28657}a^{9}-\frac{5850}{28657}a^{8}+\frac{12087}{28657}a^{7}-\frac{9865}{28657}a^{6}-\frac{3250}{28657}a^{5}-\frac{2162}{28657}a^{4}+\frac{4715}{28657}a^{3}-\frac{7078}{28657}a^{2}+\frac{568}{28657}a-\frac{11925}{28657}$, $\frac{1}{28657}a^{37}+\frac{1513}{28657}a^{23}+\frac{12112}{28657}a^{22}-\frac{8960}{28657}a^{21}+\frac{11191}{28657}a^{20}+\frac{5063}{28657}a^{19}-\frac{10949}{28657}a^{18}+\frac{5148}{28657}a^{17}+\frac{13956}{28657}a^{16}-\frac{577}{28657}a^{15}+\frac{86}{28657}a^{14}-\frac{6083}{28657}a^{13}-\frac{1612}{28657}a^{12}-\frac{3050}{28657}a^{11}-\frac{143}{28657}a^{10}-\frac{14065}{28657}a^{9}-\frac{4556}{28657}a^{8}-\frac{5751}{28657}a^{7}+\frac{335}{28657}a^{6}-\frac{10352}{28657}a^{5}-\frac{832}{28657}a^{4}+\frac{1112}{28657}a^{3}-\frac{14051}{28657}a^{2}-\frac{3737}{28657}a-\frac{8012}{28657}$, $\frac{1}{28657}a^{38}-\frac{14097}{28657}a^{23}+\frac{10709}{28657}a^{22}-\frac{13289}{28657}a^{21}-\frac{4465}{28657}a^{20}+\frac{12230}{28657}a^{19}-\frac{10578}{28657}a^{18}+\frac{4763}{28657}a^{17}+\frac{2977}{28657}a^{16}-\frac{273}{28657}a^{15}-\frac{12960}{28657}a^{14}+\frac{10336}{28657}a^{13}-\frac{213}{28657}a^{12}-\frac{13460}{28657}a^{11}+\frac{7712}{28657}a^{10}+\frac{10456}{28657}a^{9}-\frac{5575}{28657}a^{8}-\frac{14198}{28657}a^{7}-\frac{12909}{28657}a^{6}-\frac{1825}{28657}a^{5}-\frac{6898}{28657}a^{4}-\frac{13058}{28657}a^{3}+\frac{1702}{28657}a^{2}-\frac{11470}{28657}a-\frac{6052}{28657}$, $\frac{1}{28657}a^{39}-\frac{1474}{28657}a^{23}+\frac{4049}{28657}a^{22}+\frac{2737}{28657}a^{21}+\frac{3177}{28657}a^{20}-\frac{7969}{28657}a^{19}-\frac{8212}{28657}a^{18}+\frac{4724}{28657}a^{17}-\frac{752}{28657}a^{16}+\frac{7545}{28657}a^{15}-\frac{14193}{28657}a^{14}-\frac{9806}{28657}a^{13}+\frac{7856}{28657}a^{12}-\frac{13143}{28657}a^{11}-\frac{13742}{28657}a^{10}+\frac{207}{28657}a^{9}+\frac{14145}{28657}a^{8}+\frac{4878}{28657}a^{7}+\frac{12832}{28657}a^{6}-\frac{11965}{28657}a^{5}+\frac{11002}{28657}a^{4}+\frac{6769}{28657}a^{3}-\frac{9776}{28657}a^{2}+\frac{12492}{28657}a-\frac{926}{28657}$, $\frac{1}{3983323}a^{40}-\frac{43}{3983323}a^{39}-\frac{47}{3983323}a^{38}-\frac{64}{3983323}a^{37}+\frac{13}{3983323}a^{36}-\frac{3}{3983323}a^{35}-\frac{68}{3983323}a^{34}+\frac{5}{3983323}a^{33}+\frac{23}{3983323}a^{32}-\frac{16}{3983323}a^{31}+\frac{45}{3983323}a^{30}+\frac{11}{3983323}a^{29}+\frac{7}{3983323}a^{28}-\frac{23}{3983323}a^{27}-\frac{30}{3983323}a^{26}+\frac{39}{3983323}a^{25}+\frac{60}{3983323}a^{24}-\frac{400398}{3983323}a^{23}-\frac{1528952}{3983323}a^{22}-\frac{103117}{3983323}a^{21}-\frac{506271}{3983323}a^{20}+\frac{273720}{3983323}a^{19}-\frac{815018}{3983323}a^{18}+\frac{1343695}{3983323}a^{17}-\frac{811089}{3983323}a^{16}-\frac{1679061}{3983323}a^{15}-\frac{477397}{3983323}a^{14}+\frac{1538586}{3983323}a^{13}+\frac{952302}{3983323}a^{12}-\frac{1877059}{3983323}a^{11}+\frac{814551}{3983323}a^{10}+\frac{1702555}{3983323}a^{9}-\frac{144244}{3983323}a^{8}+\frac{828752}{3983323}a^{7}-\frac{1090985}{3983323}a^{6}-\frac{1526087}{3983323}a^{5}-\frac{1595416}{3983323}a^{4}+\frac{269515}{3983323}a^{3}+\frac{777450}{3983323}a^{2}-\frac{181133}{3983323}a-\frac{5034}{28657}$, $\frac{1}{3983323}a^{41}+\frac{50}{3983323}a^{39}+\frac{41}{3983323}a^{37}-\frac{58}{3983323}a^{35}-\frac{40}{3983323}a^{33}+\frac{52}{3983323}a^{31}+\frac{63}{3983323}a^{29}-\frac{46}{3983323}a^{27}+\frac{69}{3983323}a^{25}+\frac{783156}{3983323}a^{23}+\frac{9062}{28657}a^{22}+\frac{1219205}{3983323}a^{21}-\frac{7223}{28657}a^{20}+\frac{277101}{3983323}a^{19}+\frac{13744}{28657}a^{18}-\frac{582513}{3983323}a^{17}-\frac{1119}{28657}a^{16}+\frac{1008826}{3983323}a^{15}-\frac{8359}{28657}a^{14}+\frac{230260}{3983323}a^{13}-\frac{9130}{28657}a^{12}+\frac{445655}{3983323}a^{11}-\frac{4052}{28657}a^{10}-\frac{903368}{3983323}a^{9}-\frac{4140}{28657}a^{8}+\frac{839797}{3983323}a^{7}-\frac{11119}{28657}a^{6}+\frac{151417}{3983323}a^{5}+\frac{1695}{28657}a^{4}+\frac{1165558}{3983323}a^{3}-\frac{9371}{28657}a^{2}-\frac{1386657}{3983323}a-\frac{6107}{28657}$, $\frac{1}{5500969063}a^{42}+\frac{114}{5500969063}a^{41}-\frac{154}{5500969063}a^{40}-\frac{31537}{5500969063}a^{39}-\frac{94482}{5500969063}a^{38}-\frac{15074}{5500969063}a^{37}-\frac{68596}{5500969063}a^{36}+\frac{80319}{5500969063}a^{35}-\frac{10632}{5500969063}a^{34}+\frac{5123}{5500969063}a^{33}+\frac{23716}{5500969063}a^{32}-\frac{65590}{5500969063}a^{31}-\frac{43311}{5500969063}a^{30}+\frac{71936}{5500969063}a^{29}+\frac{52319}{5500969063}a^{28}-\frac{55040}{5500969063}a^{27}-\frac{67064}{5500969063}a^{26}+\frac{12698}{5500969063}a^{25}-\frac{95471}{5500969063}a^{24}+\frac{1669779628}{5500969063}a^{23}-\frac{1795757706}{5500969063}a^{22}-\frac{2028011570}{5500969063}a^{21}+\frac{2411464291}{5500969063}a^{20}+\frac{142052597}{5500969063}a^{19}+\frac{1076196489}{5500969063}a^{18}+\frac{2583475676}{5500969063}a^{17}-\frac{860725813}{5500969063}a^{16}-\frac{1504518919}{5500969063}a^{15}-\frac{2517486720}{5500969063}a^{14}-\frac{608195519}{5500969063}a^{13}-\frac{629810102}{5500969063}a^{12}+\frac{2714969632}{5500969063}a^{11}+\frac{2004505629}{5500969063}a^{10}+\frac{2101227756}{5500969063}a^{9}-\frac{269093028}{5500969063}a^{8}+\frac{570400744}{5500969063}a^{7}+\frac{2714670997}{5500969063}a^{6}+\frac{51523621}{5500969063}a^{5}-\frac{650493013}{5500969063}a^{4}-\frac{977972298}{5500969063}a^{3}+\frac{2742876477}{5500969063}a^{2}-\frac{910393118}{5500969063}a-\frac{9053653}{39575317}$, $\frac{1}{15\!\cdots\!19}a^{43}+\frac{66\!\cdots\!52}{15\!\cdots\!19}a^{42}+\frac{13\!\cdots\!34}{15\!\cdots\!19}a^{41}-\frac{31\!\cdots\!91}{15\!\cdots\!19}a^{40}+\frac{18\!\cdots\!20}{15\!\cdots\!19}a^{39}+\frac{10\!\cdots\!05}{15\!\cdots\!19}a^{38}-\frac{73\!\cdots\!94}{15\!\cdots\!19}a^{37}-\frac{10\!\cdots\!86}{15\!\cdots\!19}a^{36}+\frac{23\!\cdots\!96}{15\!\cdots\!19}a^{35}+\frac{36\!\cdots\!34}{15\!\cdots\!19}a^{34}+\frac{12\!\cdots\!82}{15\!\cdots\!19}a^{33}+\frac{19\!\cdots\!12}{15\!\cdots\!19}a^{32}+\frac{11\!\cdots\!40}{15\!\cdots\!19}a^{31}-\frac{49\!\cdots\!27}{15\!\cdots\!19}a^{30}+\frac{56\!\cdots\!40}{15\!\cdots\!19}a^{29}+\frac{26\!\cdots\!91}{15\!\cdots\!19}a^{28}-\frac{70\!\cdots\!19}{15\!\cdots\!19}a^{27}+\frac{24\!\cdots\!38}{15\!\cdots\!19}a^{26}-\frac{10\!\cdots\!50}{15\!\cdots\!19}a^{25}-\frac{70\!\cdots\!92}{15\!\cdots\!19}a^{24}-\frac{13\!\cdots\!09}{15\!\cdots\!19}a^{23}+\frac{65\!\cdots\!78}{15\!\cdots\!19}a^{22}+\frac{27\!\cdots\!41}{15\!\cdots\!19}a^{21}-\frac{47\!\cdots\!29}{15\!\cdots\!19}a^{20}-\frac{60\!\cdots\!73}{15\!\cdots\!19}a^{19}-\frac{69\!\cdots\!92}{15\!\cdots\!19}a^{18}+\frac{19\!\cdots\!20}{15\!\cdots\!19}a^{17}-\frac{19\!\cdots\!56}{15\!\cdots\!19}a^{16}+\frac{51\!\cdots\!88}{15\!\cdots\!19}a^{15}+\frac{13\!\cdots\!52}{15\!\cdots\!19}a^{14}+\frac{66\!\cdots\!44}{15\!\cdots\!19}a^{13}+\frac{39\!\cdots\!35}{15\!\cdots\!19}a^{12}-\frac{42\!\cdots\!02}{15\!\cdots\!19}a^{11}-\frac{47\!\cdots\!20}{15\!\cdots\!19}a^{10}-\frac{23\!\cdots\!62}{15\!\cdots\!19}a^{9}+\frac{56\!\cdots\!17}{15\!\cdots\!19}a^{8}-\frac{72\!\cdots\!52}{15\!\cdots\!19}a^{7}-\frac{44\!\cdots\!75}{15\!\cdots\!19}a^{6}-\frac{58\!\cdots\!56}{15\!\cdots\!19}a^{5}-\frac{76\!\cdots\!03}{15\!\cdots\!19}a^{4}-\frac{61\!\cdots\!77}{15\!\cdots\!19}a^{3}-\frac{63\!\cdots\!63}{15\!\cdots\!19}a^{2}+\frac{57\!\cdots\!44}{15\!\cdots\!19}a+\frac{34\!\cdots\!34}{24\!\cdots\!61}$ 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, 1/28657*a^24 + 10946/28657*a^23 - 13/28657*a^22 + 5168/28657*a^21 + 101/28657*a^20 - 754/28657*a^19 - 501/28657*a^18 - 7930/28657*a^17 + 1797/28657*a^16 + 11857/28657*a^15 - 4598/28657*a^14 - 3796/28657*a^13 + 8635/28657*a^12 + 9498/28657*a^11 - 11284/28657*a^10 + 14025/28657*a^9 + 10152/28657*a^8 + 12302/28657*a^7 - 5510/28657*a^6 - 6477/28657*a^5 + 1691/28657*a^4 + 6477/28657*a^3 - 193/28657*a^2 + 11556/28657*a + 4/28657, 1/28657*a^25 - 12/28657*a^23 + 4181/28657*a^22 + 91/28657*a^21 + 11323/28657*a^20 - 433/28657*a^19 + 2529/28657*a^18 + 1524/28657*a^17 + 597/28657*a^16 - 3767/28657*a^15 + 4220/28657*a^14 + 7001/28657*a^13 + 1574/28657*a^12 - 8796/28657*a^11 - 11638/28657*a^10 + 8051/28657*a^9 - 8301/28657*a^8 - 3959/28657*a^7 + 11655/28657*a^6 + 1515/28657*a^5 + 9213/28657*a^4 - 17/28657*a^3 + 3516/28657*a^2 + 26/28657*a + 13530/28657, 1/28657*a^26 - 7752/28657*a^23 - 65/28657*a^22 - 12632/28657*a^21 + 779/28657*a^20 - 6519/28657*a^19 - 4488/28657*a^18 - 8592/28657*a^17 - 10860/28657*a^16 + 3219/28657*a^15 + 9139/28657*a^14 + 13336/28657*a^13 + 8853/28657*a^12 - 12290/28657*a^11 - 12729/28657*a^10 - 11943/28657*a^9 + 3237/28657*a^8 - 12663/28657*a^7 - 7291/28657*a^6 - 11197/28657*a^5 - 8382/28657*a^4 - 4731/28657*a^3 - 2290/28657*a^2 + 8917/28657*a + 48/28657, 1/28657*a^27 - 50/28657*a^23 + 1220/28657*a^22 + 629/28657*a^21 + 2694/28657*a^20 - 3468/28657*a^19 + 5008/28657*a^18 + 13702/28657*a^17 + 6261/28657*a^16 - 7053/28657*a^15 - 9709/28657*a^14 + 13000/28657*a^13 + 12135/28657*a^12 - 4066/28657*a^11 + 4310/28657*a^10 + 379/28657*a^9 - 6481/28657*a^8 - 12683/28657*a^7 + 2870/28657*a^6 - 11022/28657*a^5 + 7652/28657*a^4 + 350/28657*a^3 + 2945/28657*a^2 + 378/28657*a + 2351/28657, 1/28657*a^28 + 4037/28657*a^23 - 21/28657*a^22 + 3181/28657*a^21 + 1582/28657*a^20 - 4035/28657*a^19 - 11348/28657*a^18 + 10959/28657*a^17 - 3174/28657*a^16 + 10001/28657*a^15 + 12356/28657*a^14 - 5723/28657*a^13 - 2171/28657*a^12 - 7959/28657*a^11 + 9319/28657*a^10 + 7001/28657*a^9 + 7748/28657*a^8 - 12484/28657*a^7 + 48/28657*a^6 - 971/28657*a^5 - 1071/28657*a^4 + 11568/28657*a^3 - 9272/28657*a^2 + 7011/28657*a + 200/28657, 1/28657*a^29 + 71/28657*a^23 - 1652/28657*a^22 + 662/28657*a^21 - 10574/28657*a^20 - 5092/28657*a^19 - 1151/28657*a^18 + 367/28657*a^17 + 5733/28657*a^16 + 2837/28657*a^15 - 13333/28657*a^14 - 9214/28657*a^13 + 8115/28657*a^12 + 8959/28657*a^11 - 4121/28657*a^10 - 13602/28657*a^9 + 12059/28657*a^8 - 545/28657*a^7 + 5067/28657*a^6 + 11394/28657*a^5 + 5367/28657*a^4 + 6920/28657*a^3 + 12413/28657*a^2 + 2224/28657*a + 12509/28657, 1/28657*a^30 - 5079/28657*a^23 + 1585/28657*a^22 - 4961/28657*a^21 - 12263/28657*a^20 - 4931/28657*a^19 + 7281/28657*a^18 - 4377/28657*a^17 - 10122/28657*a^16 + 4530/28657*a^15 + 2017/28657*a^14 - 8939/28657*a^13 - 2329/28657*a^12 + 9289/28657*a^11 + 13823/28657*a^10 - 9378/28657*a^9 - 4912/28657*a^8 - 8665/28657*a^7 + 1406/28657*a^6 + 6722/28657*a^5 + 1487/28657*a^4 + 11058/28657*a^3 - 12730/28657*a^2 - 5571/28657*a - 284/28657, 1/28657*a^31 + 1739/28657*a^23 - 13674/28657*a^22 - 13803/28657*a^21 - 7778/28657*a^20 - 10904/28657*a^19 + 1517/28657*a^18 + 5150/28657*a^17 - 10090/28657*a^16 - 13294/28657*a^15 - 6726/28657*a^14 + 3948/28657*a^13 - 7413/28657*a^12 - 4223/28657*a^11 - 6814/28657*a^10 - 13239/28657*a^9 - 600/28657*a^8 + 11004/28657*a^7 - 9336/28657*a^6 + 3040/28657*a^5 + 2547/28657*a^4 - 14283/28657*a^3 - 11480/28657*a^2 + 3104/28657*a - 8341/28657, 1/28657*a^32 + 8137/28657*a^23 + 8804/28657*a^22 + 3368/28657*a^21 + 14056/28657*a^20 - 5499/28657*a^19 - 11978/28657*a^18 - 3837/28657*a^17 + 13993/28657*a^16 + 6991/28657*a^15 + 4567/28657*a^14 + 2721/28657*a^13 - 4220/28657*a^12 + 11253/28657*a^11 + 8249/28657*a^10 - 2968/28657*a^9 + 9388/28657*a^8 + 4265/28657*a^7 + 13492/28657*a^6 + 3849/28657*a^5 - 3261/28657*a^4 - 12782/28657*a^3 - 5153/28657*a^2 + 12989/28657*a - 6956/28657, 1/28657*a^33 + 7158/28657*a^23 - 5479/28657*a^22 + 1859/28657*a^21 + 3717/28657*a^20 - 9278/28657*a^19 + 3506/28657*a^18 + 4839/28657*a^17 - 128/28657*a^16 + 12277/28657*a^15 - 9395/28657*a^14 - 8414/28657*a^13 - 13435/28657*a^12 + 10952/28657*a^11 - 2088/28657*a^10 + 137/28657*a^9 - 13085/28657*a^8 + 11019/28657*a^7 - 9486/28657*a^6 - 135/28657*a^5 + 11568/28657*a^4 - 8279/28657*a^3 + 7295/28657*a^2 + 14146/28657*a - 3891/28657, 1/28657*a^34 - 8709/28657*a^23 + 8942/28657*a^22 + 7360/28657*a^21 + 12846/28657*a^20 + 13122/28657*a^19 + 8872/28657*a^18 - 6705/28657*a^17 - 12313/28657*a^16 + 233/28657*a^15 + 5834/28657*a^14 - 8503/28657*a^13 - 13886/28657*a^12 + 14289/28657*a^11 - 13074/28657*a^10 + 10093/28657*a^9 - 11502/28657*a^8 - 4241/28657*a^7 + 8413/28657*a^6 + 6908/28657*a^5 + 9454/28657*a^4 + 11955/28657*a^3 - 8553/28657*a^2 + 11020/28657*a + 25/28657, 1/28657*a^35 - 4183/28657*a^23 + 8771/28657*a^22 + 811/28657*a^21 + 4364/28657*a^20 + 4739/28657*a^19 - 14050/28657*a^18 - 11313/28657*a^17 + 3584/28657*a^16 - 11381/28657*a^15 + 10001/28657*a^14 - 3072/28657*a^13 - 8121/28657*a^12 + 906/28657*a^11 + 2590/28657*a^10 - 3911/28657*a^9 + 2682/28657*a^8 - 1992/28657*a^7 - 7864/28657*a^6 - 1763/28657*a^5 + 9176/28657*a^4 + 2664/28657*a^3 - 7711/28657*a^2 - 2155/28657*a + 6179/28657, 1/28657*a^36 + 2003/28657*a^23 + 3746/28657*a^22 - 13927/28657*a^21 - 2633/28657*a^20 + 12895/28657*a^19 + 13622/28657*a^18 - 11457/28657*a^17 - 2664/28657*a^16 + 2565/28657*a^15 - 7659/28657*a^14 - 10811/28657*a^13 + 13291/28657*a^12 + 14122/28657*a^11 - 6804/28657*a^10 + 8378/28657*a^9 - 5850/28657*a^8 + 12087/28657*a^7 - 9865/28657*a^6 - 3250/28657*a^5 - 2162/28657*a^4 + 4715/28657*a^3 - 7078/28657*a^2 + 568/28657*a - 11925/28657, 1/28657*a^37 + 1513/28657*a^23 + 12112/28657*a^22 - 8960/28657*a^21 + 11191/28657*a^20 + 5063/28657*a^19 - 10949/28657*a^18 + 5148/28657*a^17 + 13956/28657*a^16 - 577/28657*a^15 + 86/28657*a^14 - 6083/28657*a^13 - 1612/28657*a^12 - 3050/28657*a^11 - 143/28657*a^10 - 14065/28657*a^9 - 4556/28657*a^8 - 5751/28657*a^7 + 335/28657*a^6 - 10352/28657*a^5 - 832/28657*a^4 + 1112/28657*a^3 - 14051/28657*a^2 - 3737/28657*a - 8012/28657, 1/28657*a^38 - 14097/28657*a^23 + 10709/28657*a^22 - 13289/28657*a^21 - 4465/28657*a^20 + 12230/28657*a^19 - 10578/28657*a^18 + 4763/28657*a^17 + 2977/28657*a^16 - 273/28657*a^15 - 12960/28657*a^14 + 10336/28657*a^13 - 213/28657*a^12 - 13460/28657*a^11 + 7712/28657*a^10 + 10456/28657*a^9 - 5575/28657*a^8 - 14198/28657*a^7 - 12909/28657*a^6 - 1825/28657*a^5 - 6898/28657*a^4 - 13058/28657*a^3 + 1702/28657*a^2 - 11470/28657*a - 6052/28657, 1/28657*a^39 - 1474/28657*a^23 + 4049/28657*a^22 + 2737/28657*a^21 + 3177/28657*a^20 - 7969/28657*a^19 - 8212/28657*a^18 + 4724/28657*a^17 - 752/28657*a^16 + 7545/28657*a^15 - 14193/28657*a^14 - 9806/28657*a^13 + 7856/28657*a^12 - 13143/28657*a^11 - 13742/28657*a^10 + 207/28657*a^9 + 14145/28657*a^8 + 4878/28657*a^7 + 12832/28657*a^6 - 11965/28657*a^5 + 11002/28657*a^4 + 6769/28657*a^3 - 9776/28657*a^2 + 12492/28657*a - 926/28657, 1/3983323*a^40 - 43/3983323*a^39 - 47/3983323*a^38 - 64/3983323*a^37 + 13/3983323*a^36 - 3/3983323*a^35 - 68/3983323*a^34 + 5/3983323*a^33 + 23/3983323*a^32 - 16/3983323*a^31 + 45/3983323*a^30 + 11/3983323*a^29 + 7/3983323*a^28 - 23/3983323*a^27 - 30/3983323*a^26 + 39/3983323*a^25 + 60/3983323*a^24 - 400398/3983323*a^23 - 1528952/3983323*a^22 - 103117/3983323*a^21 - 506271/3983323*a^20 + 273720/3983323*a^19 - 815018/3983323*a^18 + 1343695/3983323*a^17 - 811089/3983323*a^16 - 1679061/3983323*a^15 - 477397/3983323*a^14 + 1538586/3983323*a^13 + 952302/3983323*a^12 - 1877059/3983323*a^11 + 814551/3983323*a^10 + 1702555/3983323*a^9 - 144244/3983323*a^8 + 828752/3983323*a^7 - 1090985/3983323*a^6 - 1526087/3983323*a^5 - 1595416/3983323*a^4 + 269515/3983323*a^3 + 777450/3983323*a^2 - 181133/3983323*a - 5034/28657, 1/3983323*a^41 + 50/3983323*a^39 + 41/3983323*a^37 - 58/3983323*a^35 - 40/3983323*a^33 + 52/3983323*a^31 + 63/3983323*a^29 - 46/3983323*a^27 + 69/3983323*a^25 + 783156/3983323*a^23 + 9062/28657*a^22 + 1219205/3983323*a^21 - 7223/28657*a^20 + 277101/3983323*a^19 + 13744/28657*a^18 - 582513/3983323*a^17 - 1119/28657*a^16 + 1008826/3983323*a^15 - 8359/28657*a^14 + 230260/3983323*a^13 - 9130/28657*a^12 + 445655/3983323*a^11 - 4052/28657*a^10 - 903368/3983323*a^9 - 4140/28657*a^8 + 839797/3983323*a^7 - 11119/28657*a^6 + 151417/3983323*a^5 + 1695/28657*a^4 + 1165558/3983323*a^3 - 9371/28657*a^2 - 1386657/3983323*a - 6107/28657, 1/5500969063*a^42 + 114/5500969063*a^41 - 154/5500969063*a^40 - 31537/5500969063*a^39 - 94482/5500969063*a^38 - 15074/5500969063*a^37 - 68596/5500969063*a^36 + 80319/5500969063*a^35 - 10632/5500969063*a^34 + 5123/5500969063*a^33 + 23716/5500969063*a^32 - 65590/5500969063*a^31 - 43311/5500969063*a^30 + 71936/5500969063*a^29 + 52319/5500969063*a^28 - 55040/5500969063*a^27 - 67064/5500969063*a^26 + 12698/5500969063*a^25 - 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4419798294203972439419435576882670856451756706873381679452497291689103427728014031953202488446229440764850726334448014673525037273819330475/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a^6 - 5851451037656635487677186863512485634298926323188500772369420017808954542122259879361036026254721545732080877269534734297717220299249979056/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a^5 - 7613699376052171047933712058876363527692207956860374674006687390935780474254983967799000361123531509977444256175850711303360481492924663203/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a^4 - 6120928229035696784906806667948506390966881878570783560717862658396840479848282556019573296087472032208692588788559753523698154073495729677/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a^3 - 6323068115447294576202595885252326284829375901788296557332647111885185083996183579500752112267604354220404382948881994800656877850069933063/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a^2 + 5717505891007021129464956519660029337646096002656557246475428173312658074144547750873730195787822360294062995690831355087340263684361439544/15548871779422679760733924651630165416533823791251887956811700137031266722304982404895289485804787291799575316024861565336925547503186431619*a + 34926502859605264975420647184272472266842827014345368462986190711554966783269874432391543727811980632067469208582736868335721662669534/242651598486597477500178290104873131861199828200375910310892806333295880433605118758021964852834583745058058272208704338971044296933261

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)

Relative class number: data not computed

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{30396202823626472809883862808913371608380021398424097524446400111803106595028279803557138639319633495530237165205872335129972}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{43} - \frac{16244059232468368819987914929694039806780558468851032754244454209648431429251629725845218144259362961200440049337307929888800}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{42} + \frac{968137064383691479455797790027366347478663284438929571650112883475526246919001940066012149891662258369411027161661740874094087}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{41} + \frac{284122481947603482556560960554884176187921625732525718680452049622648738230216359699266612936093171283406637978849786878228258}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{40} - \frac{14883588320661136659929700316490985428854579067225418325980625740019895978144572184835543089775237610359235308951150049410522859}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{39} - \frac{2462743070832765499617484195363019906262497395764846070892184381067966960650020706656551067152279969688802486999936456092964351}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{38} + \frac{1104575138588386186827193405875070672891618309240865751615711524130564373169868176855600885141759848674152108363983453498961016}{3903492581300256032647597157355847220156091733271915924671863199904016501374601659191406141506673521529530825400014401377173166098553} a^{37} + \frac{9571995544119863113726595260203225772324307488523262391189013931447753615447919392396936867407206613996299905554999844319760564}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{36} - \frac{1169970746852433326510935455880975723698005903073657939567948854056750930053535537166329904931749145569333888278024177117509265252}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{35} + \frac{31670898463367378090005780259162813051189868217223692691344080344872159253480862427039321527039904724070242911635551034163561209}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{34} + \frac{6996152673033287308700272995699466578326382458540889679724224204348258400144740567499402240207644002960359231160365230874935595571}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{33} - \frac{772437944830730513728819439271512979672367038243367244593188527328440724072927186150439019520472883197531142313657787875496139637}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{32} - \frac{33751144127957791376408336763214066199943378956596995109694931442343538641956560213269703544691198131854774188773656989706994314144}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{31} + \frac{6490338193883388350265965265857157785810555023445301815181929558856092572960658263361951407218833847207084096897211952106255862350}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{30} + \frac{134216350151836949377167596351755171737354381595125831992299036627946432241054240247154408978208082470670414546235712828676482374079}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{29} - \frac{36898831737542735710660465127603559239833584050939901625652609577704930438972880323184008551536056021210908595968056324466852038138}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{28} - \frac{443968286639389537687370538990850854941011795436105648613314885021585642208080929985591060644051470513715432789549536144845246629026}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{27} + \frac{160726411830574013792284855229708953347044942619661139309527621850180982725173067621408920838913949230131789272371856538871088441057}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{26} + \frac{1231933711184688417933492721837288724461445020220142242305151823444742019543667762620235057667050551853290560863749463704130852747875}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{25} - \frac{559909475811860824918017126765600716557920246969181757270720922850269024976144405888986980053518222548616488607626206070646765096040}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{24} - \frac{2868372626328137287741484936525240045737558568718016880276543514547368194996568054739630323321581826847280822510674240193600539687383}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{23} + \frac{1597012828417528991319614848312256354444111305822888928851514712055463031887532253632904577608088809194476347074868853262338092735613}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{22} + \frac{5624245425351843116992250953981742542442663811751689706605543680232778442369209766466230897925459316806040196185237973186167222380399}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{21} - \frac{3750467672988327205397646287351771740715980756666237767773197644018820936909966645334633714429796540793806152795295034491363470163978}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{20} - \frac{9254132208484057015290670213101617197631380875258143405485413283775368899291077259149452678474569365585604269853132632187960788580301}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{19} + \frac{7388166564284404154920579974771904248766662068553483294119660434590435792826029538443485541862338196573890655813377708325805588765174}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{18} + \frac{12985575172140954063708974669388355868004903096367656276893736372401966157132866574592995343920431340060727738097189364649746216263721}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{17} - \frac{12747746011507601680484551317941272004722197828793057556326359015640557764248399796756957815976427250696472181277859584546970687792603}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{16} - \frac{15506221765282796773733162148250114323158253476242987300888111561135837315442735260693992388644830590615550400610247771031190198854296}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{15} + \frac{15229536539553680463773706482699717647302704328176202426392833225450979822163671005991692011054238179087716909870031097983190791318820}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{14} + \frac{13922693004865941537839117779306028367678183787869340853522632851762262835905254719758966772149651396265801938489430795657012221834105}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{13} + \frac{29094700593615157317413103488766831379359986909155660694482678736300245006766123308764878637604319961054370995371425169742428909778985}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{12} - \frac{14131180285463567132638421687239915761091284302438857022198871718933769093115175039461099364872061415336967970361830876304247160904735}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{11} - \frac{130829983360154711532651293095460741229664091380445371146933513828650748909552601688795122817977293397727606392269100996657682424746686}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{10} + \frac{100086180798998278912854405561416875111966005683184860111957476119693861203424705490420934211564073833168087477430371919012130902355787}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{9} + \frac{23177173636597789063224518273870427668865499690994442966484673752878068206617690528206198912511356252697998130643697107232718136431335}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{8} - \frac{235576122257433666413887440259554852617359443937000754859662265714092516605708456857127358935014352923469748574082351467374623225266679}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{7} + \frac{327931366952256047738672216783892350953897830632542422738668819415741268520060454029566605695532979998135378303103112169782099131034938}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{6} + \frac{18836082930136174628636166387263271362527159077828686429710244491002347579984769108318817394798930828919986193283143698175872185246745}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{5} - \frac{438423885413599488414134181834383024435638706364497351381923530629376031522198619928034146620074586512323331916792344961812018480407259}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{4} + \frac{386886777460609735767238860245816411646468192718569229105292590785449723458023611276698815968243196931662995846780928056271625705779273}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{3} - \frac{96628510794923113424672257609959036731096430588468088795150846541105134619079078059603977821041921555150538186797206610715500793257744}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{2} - \frac{513093405889685415836534827638314283809859316346556566268285086476198000973543445336233066970142955379745039964017055465268694534211811}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a + \frac{10517775915641457695712169198631401228854761233558600729390777724711416516133785463229438130567557111588750896862676065546352633660}{8467445946421379680363551317474722820295209833561639749830505856624764645064211842063787725611005469695294632104152714484106650973} \) -(30396202823626472809883862808913371608380021398424097524446400111803106595028279803557138639319633495530237165205872335129972)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(43) - 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(36898831737542735710660465127603559239833584050939901625652609577704930438972880323184008551536056021210908595968056324466852038138)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(28) - (443968286639389537687370538990850854941011795436105648613314885021585642208080929985591060644051470513715432789549536144845246629026)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(27) + (160726411830574013792284855229708953347044942619661139309527621850180982725173067621408920838913949230131789272371856538871088441057)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(26) + (1231933711184688417933492721837288724461445020220142242305151823444742019543667762620235057667050551853290560863749463704130852747875)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(25) - (559909475811860824918017126765600716557920246969181757270720922850269024976144405888986980053518222548616488607626206070646765096040)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(24) - (2868372626328137287741484936525240045737558568718016880276543514547368194996568054739630323321581826847280822510674240193600539687383)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(23) + (1597012828417528991319614848312256354444111305822888928851514712055463031887532253632904577608088809194476347074868853262338092735613)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(22) + (5624245425351843116992250953981742542442663811751689706605543680232778442369209766466230897925459316806040196185237973186167222380399)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(21) - (3750467672988327205397646287351771740715980756666237767773197644018820936909966645334633714429796540793806152795295034491363470163978)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(20) - (9254132208484057015290670213101617197631380875258143405485413283775368899291077259149452678474569365585604269853132632187960788580301)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(19) + (7388166564284404154920579974771904248766662068553483294119660434590435792826029538443485541862338196573890655813377708325805588765174)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(18) + (12985575172140954063708974669388355868004903096367656276893736372401966157132866574592995343920431340060727738097189364649746216263721)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(17) - (12747746011507601680484551317941272004722197828793057556326359015640557764248399796756957815976427250696472181277859584546970687792603)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(16) - (15506221765282796773733162148250114323158253476242987300888111561135837315442735260693992388644830590615550400610247771031190198854296)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(15) + (15229536539553680463773706482699717647302704328176202426392833225450979822163671005991692011054238179087716909870031097983190791318820)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(14) + (13922693004865941537839117779306028367678183787869340853522632851762262835905254719758966772149651396265801938489430795657012221834105)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(13) + (29094700593615157317413103488766831379359986909155660694482678736300245006766123308764878637604319961054370995371425169742428909778985)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(12) - (14131180285463567132638421687239915761091284302438857022198871718933769093115175039461099364872061415336967970361830876304247160904735)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(11) - (130829983360154711532651293095460741229664091380445371146933513828650748909552601688795122817977293397727606392269100996657682424746686)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(10) + (100086180798998278912854405561416875111966005683184860111957476119693861203424705490420934211564073833168087477430371919012130902355787)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(9) + (23177173636597789063224518273870427668865499690994442966484673752878068206617690528206198912511356252697998130643697107232718136431335)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(8) - (235576122257433666413887440259554852617359443937000754859662265714092516605708456857127358935014352923469748574082351467374623225266679)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(7) + (327931366952256047738672216783892350953897830632542422738668819415741268520060454029566605695532979998135378303103112169782099131034938)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(6) + (18836082930136174628636166387263271362527159077828686429710244491002347579984769108318817394798930828919986193283143698175872185246745)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(5) - (438423885413599488414134181834383024435638706364497351381923530629376031522198619928034146620074586512323331916792344961812018480407259)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(4) + (386886777460609735767238860245816411646468192718569229105292590785449723458023611276698815968243196931662995846780928056271625705779273)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(3) - (96628510794923113424672257609959036731096430588468088795150846541105134619079078059603977821041921555150538186797206610715500793257744)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)a^(2) - (513093405889685415836534827638314283809859316346556566268285086476198000973543445336233066970142955379745039964017055465268694534211811)/(542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867)*a + (10517775915641457695712169198631401228854761233558600729390777724711416516133785463229438130567557111588750896862676065546352633660)/(8467445946421379680363551317474722820295209833561639749830505856624764645064211842063787725611005469695294632104152714484106650973) (order $6$)	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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{22}\cdot R \cdot h}{6\cdot\sqrt{116567320065927752512435466812933331534234135648947894549180382317450046539306640625}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)

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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241, 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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);

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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);

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{345}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-115}) $, $\Q(\sqrt{-3}, \sqrt{-115})$, $\Q(\zeta_{23})^+$, 22.22.341419566026798986253349758444608447265625.1, 22.0.304011857053427966889939263171547.1, 22.0.1927323443393334271838358868310546875.1

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	$22^{2}$	R	R	${\href{/padicField/7.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	R	$22^{2}$	${\href{/padicField/31.11.0.1}{11} }^{4}$	${\href{/padicField/37.11.0.1}{11} }^{4}$	$22^{2}$	${\href{/padicField/43.11.0.1}{11} }^{4}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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

$p$	Polynomial	$e$	$f$	$c$
$3$ 3	Deg $44$	$2$	$22$	$22$
$5$ 5	Deg $44$	$2$	$22$	$22$
$23$ 23	Deg $44$	$22$	$2$	$42$

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