LMFDB - Number field 32.0.138956997215838851269528412416000000000000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561)

gp: K = bnfinit(y^32 + 3*y^30 - 9*y^28 - 85*y^26 - 114*y^24 - 1521*y^22 - 2090*y^20 + 22329*y^18 + 111717*y^16 - 44722*y^14 - 41235*y^12 + 33462*y^10 + 16051*y^8 + 13275*y^6 - 11826*y^4 - 2916*y^2 + 6561, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561)

$ x^{32} + 3 x^{30} - 9 x^{28} - 85 x^{26} - 114 x^{24} - 1521 x^{22} - 2090 x^{20} + 22329 x^{18} + \cdots + 6561 $ x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$138956997215838851269528412416000000000000000000000000$ 138956997215838851269528412416000000000000000000000000 $\medspace = 2^{32}\cdot 5^{24}\cdot 13^{24}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$45.78$	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:	$2\cdot 5^{3/4}13^{3/4}\approx 45.78413496570207$
Ramified primes:	$2$, $5$, $13$ 2, 5, 13	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$32$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$260=2^{2}\cdot 5\cdot 13$
Dirichlet character group:	$\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(259,·)$, $\chi_{260}(129,·)$, $\chi_{260}(131,·)$, $\chi_{260}(21,·)$, $\chi_{260}(151,·)$, $\chi_{260}(27,·)$, $\chi_{260}(157,·)$, $\chi_{260}(31,·)$, $\chi_{260}(161,·)$, $\chi_{260}(47,·)$, $\chi_{260}(177,·)$, $\chi_{260}(51,·)$, $\chi_{260}(53,·)$, $\chi_{260}(183,·)$, $\chi_{260}(57,·)$, $\chi_{260}(187,·)$, $\chi_{260}(181,·)$, $\chi_{260}(73,·)$, $\chi_{260}(203,·)$, $\chi_{260}(77,·)$, $\chi_{260}(79,·)$, $\chi_{260}(209,·)$, $\chi_{260}(83,·)$, $\chi_{260}(213,·)$, $\chi_{260}(207,·)$, $\chi_{260}(99,·)$, $\chi_{260}(229,·)$, $\chi_{260}(103,·)$, $\chi_{260}(233,·)$, $\chi_{260}(109,·)$, $\chi_{260}(239,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{5}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{6}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{7}$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{16}-\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{2}{9}a^{8}+\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{19}-\frac{2}{27}a^{17}+\frac{1}{9}a^{15}-\frac{4}{27}a^{13}-\frac{4}{27}a^{11}-\frac{4}{27}a^{9}+\frac{4}{9}a^{7}-\frac{11}{27}a^{5}+\frac{1}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{108}a^{20}+\frac{1}{27}a^{18}-\frac{1}{27}a^{14}+\frac{2}{27}a^{12}-\frac{1}{108}a^{10}-\frac{1}{9}a^{8}+\frac{4}{27}a^{6}+\frac{13}{27}a^{4}+\frac{2}{9}a^{2}-\frac{1}{4}$, $\frac{1}{108}a^{21}+\frac{2}{27}a^{17}-\frac{4}{27}a^{15}-\frac{1}{9}a^{13}+\frac{5}{36}a^{11}+\frac{1}{27}a^{9}-\frac{8}{27}a^{7}-\frac{1}{9}a^{5}-\frac{4}{27}a^{3}+\frac{5}{12}a$, $\frac{1}{108}a^{22}-\frac{1}{27}a^{18}+\frac{2}{27}a^{16}-\frac{1}{9}a^{14}-\frac{1}{12}a^{12}+\frac{4}{27}a^{10}+\frac{13}{27}a^{8}+\frac{2}{9}a^{6}-\frac{7}{27}a^{4}+\frac{11}{36}a^{2}$, $\frac{1}{108}a^{23}+\frac{11}{108}a^{13}+\frac{1}{108}a^{3}$, $\frac{1}{108}a^{24}+\frac{11}{108}a^{14}+\frac{1}{108}a^{4}$, $\frac{1}{108}a^{25}+\frac{11}{108}a^{15}+\frac{1}{108}a^{5}$, $\frac{1}{1456295763348}a^{26}+\frac{436656233}{242715960558}a^{24}-\frac{8598646}{40452660093}a^{22}+\frac{162482033}{40452660093}a^{20}+\frac{491600858}{13484220031}a^{18}+\frac{123874714055}{1456295763348}a^{16}-\frac{18370971659}{242715960558}a^{14}+\frac{5004295789}{40452660093}a^{12}+\frac{1877518436}{13484220031}a^{10}+\frac{14849052893}{40452660093}a^{8}-\frac{229642858259}{1456295763348}a^{6}+\frac{115372803131}{242715960558}a^{4}+\frac{1952193478}{13484220031}a^{2}-\frac{1226994442}{13484220031}$, $\frac{1}{1456295763348}a^{27}+\frac{436656233}{242715960558}a^{25}-\frac{8598646}{40452660093}a^{23}+\frac{162482033}{40452660093}a^{21}-\frac{210996865}{364073940837}a^{19}+\frac{231748474303}{1456295763348}a^{17}+\frac{35565908465}{242715960558}a^{15}-\frac{22382438054}{364073940837}a^{13}-\frac{16728102383}{364073940837}a^{11}-\frac{55137604397}{364073940837}a^{9}+\frac{93978422485}{1456295763348}a^{7}-\frac{328092592157}{728147881674}a^{5}+\frac{160582984154}{364073940837}a^{3}-\frac{17165203357}{40452660093}a$, $\frac{1}{4368887290044}a^{28}-\frac{756717713}{1456295763348}a^{24}-\frac{8397050963}{2184443645022}a^{22}-\frac{3173183095}{1456295763348}a^{20}+\frac{5400358661}{161810640372}a^{18}-\frac{168127493225}{1092221822511}a^{16}-\frac{171885519311}{1456295763348}a^{14}+\frac{1736565031}{728147881674}a^{12}+\frac{496758606893}{4368887290044}a^{10}-\frac{260557156309}{1456295763348}a^{8}-\frac{41468490176}{121357980279}a^{6}-\frac{1786851985187}{4368887290044}a^{4}+\frac{17199500173}{242715960558}a^{2}+\frac{11770936847}{53936880124}$, $\frac{1}{13106661870132}a^{29}+\frac{1}{4368887290044}a^{27}-\frac{1936833391}{728147881674}a^{25}-\frac{58175415787}{13106661870132}a^{23}-\frac{5404024969}{2184443645022}a^{21}-\frac{20038173253}{1456295763348}a^{19}+\frac{346356730753}{13106661870132}a^{17}+\frac{45123612001}{728147881674}a^{15}+\frac{119640879623}{1456295763348}a^{13}+\frac{491858299939}{6553330935066}a^{11}-\frac{588981334145}{4368887290044}a^{9}-\frac{86800180055}{485431921116}a^{7}-\frac{2059740739483}{6553330935066}a^{5}+\frac{147261155443}{485431921116}a^{3}-\frac{16794850507}{80905320186}a$, $\frac{1}{39319985610396}a^{30}+\frac{1}{13106661870132}a^{28}-\frac{1}{4368887290044}a^{26}+\frac{9257801039}{9829996402599}a^{24}-\frac{29349668189}{13106661870132}a^{22}+\frac{11712482075}{4368887290044}a^{20}+\frac{2117068222525}{39319985610396}a^{18}-\frac{17551756493}{161810640372}a^{16}+\frac{89087782901}{1092221822511}a^{14}-\frac{5001641332921}{39319985610396}a^{12}-\frac{706711017953}{13106661870132}a^{10}-\frac{1981734423185}{4368887290044}a^{8}-\frac{15464536204685}{39319985610396}a^{6}+\frac{324155339690}{1092221822511}a^{4}+\frac{73440071909}{485431921116}a^{2}-\frac{5866965635}{13484220031}$, $\frac{1}{117959956831188}a^{31}+\frac{1}{39319985610396}a^{29}-\frac{1}{13106661870132}a^{27}+\frac{9257801039}{29489989207797}a^{25}+\frac{46004156045}{19659992805198}a^{23}+\frac{11712482075}{13106661870132}a^{21}+\frac{660772459177}{117959956831188}a^{19}-\frac{535523969305}{4368887290044}a^{17}-\frac{32270197378}{3276665467533}a^{15}+\frac{2414177534839}{58979978415594}a^{13}+\frac{1235016666511}{39319985610396}a^{11}+\frac{121803901651}{13106661870132}a^{9}-\frac{19833423494729}{117959956831188}a^{7}+\frac{1133208541550}{3276665467533}a^{5}+\frac{191788566311}{728147881674}a^{3}+\frac{7617254396}{40452660093}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 - 1/3*a^7 + 1/3*a^5 - 1/3*a^3 + 1/3*a, 1/3*a^10 - 1/3*a^8 + 1/3*a^6 - 1/3*a^4 + 1/3*a^2, 1/3*a^11 + 1/3*a, 1/3*a^12 + 1/3*a^2, 1/3*a^13 + 1/3*a^3, 1/3*a^14 + 1/3*a^4, 1/3*a^15 + 1/3*a^5, 1/3*a^16 + 1/3*a^6, 1/3*a^17 + 1/3*a^7, 1/9*a^18 + 1/9*a^16 - 1/9*a^12 - 1/9*a^10 + 2/9*a^8 + 1/9*a^4 - 2/9*a^2, 1/27*a^19 - 2/27*a^17 + 1/9*a^15 - 4/27*a^13 - 4/27*a^11 - 4/27*a^9 + 4/9*a^7 - 11/27*a^5 + 1/27*a^3 + 1/3*a, 1/108*a^20 + 1/27*a^18 - 1/27*a^14 + 2/27*a^12 - 1/108*a^10 - 1/9*a^8 + 4/27*a^6 + 13/27*a^4 + 2/9*a^2 - 1/4, 1/108*a^21 + 2/27*a^17 - 4/27*a^15 - 1/9*a^13 + 5/36*a^11 + 1/27*a^9 - 8/27*a^7 - 1/9*a^5 - 4/27*a^3 + 5/12*a, 1/108*a^22 - 1/27*a^18 + 2/27*a^16 - 1/9*a^14 - 1/12*a^12 + 4/27*a^10 + 13/27*a^8 + 2/9*a^6 - 7/27*a^4 + 11/36*a^2, 1/108*a^23 + 11/108*a^13 + 1/108*a^3, 1/108*a^24 + 11/108*a^14 + 1/108*a^4, 1/108*a^25 + 11/108*a^15 + 1/108*a^5, 1/1456295763348*a^26 + 436656233/242715960558*a^24 - 8598646/40452660093*a^22 + 162482033/40452660093*a^20 + 491600858/13484220031*a^18 + 123874714055/1456295763348*a^16 - 18370971659/242715960558*a^14 + 5004295789/40452660093*a^12 + 1877518436/13484220031*a^10 + 14849052893/40452660093*a^8 - 229642858259/1456295763348*a^6 + 115372803131/242715960558*a^4 + 1952193478/13484220031*a^2 - 1226994442/13484220031, 1/1456295763348*a^27 + 436656233/242715960558*a^25 - 8598646/40452660093*a^23 + 162482033/40452660093*a^21 - 210996865/364073940837*a^19 + 231748474303/1456295763348*a^17 + 35565908465/242715960558*a^15 - 22382438054/364073940837*a^13 - 16728102383/364073940837*a^11 - 55137604397/364073940837*a^9 + 93978422485/1456295763348*a^7 - 328092592157/728147881674*a^5 + 160582984154/364073940837*a^3 - 17165203357/40452660093*a, 1/4368887290044*a^28 - 756717713/1456295763348*a^24 - 8397050963/2184443645022*a^22 - 3173183095/1456295763348*a^20 + 5400358661/161810640372*a^18 - 168127493225/1092221822511*a^16 - 171885519311/1456295763348*a^14 + 1736565031/728147881674*a^12 + 496758606893/4368887290044*a^10 - 260557156309/1456295763348*a^8 - 41468490176/121357980279*a^6 - 1786851985187/4368887290044*a^4 + 17199500173/242715960558*a^2 + 11770936847/53936880124, 1/13106661870132*a^29 + 1/4368887290044*a^27 - 1936833391/728147881674*a^25 - 58175415787/13106661870132*a^23 - 5404024969/2184443645022*a^21 - 20038173253/1456295763348*a^19 + 346356730753/13106661870132*a^17 + 45123612001/728147881674*a^15 + 119640879623/1456295763348*a^13 + 491858299939/6553330935066*a^11 - 588981334145/4368887290044*a^9 - 86800180055/485431921116*a^7 - 2059740739483/6553330935066*a^5 + 147261155443/485431921116*a^3 - 16794850507/80905320186*a, 1/39319985610396*a^30 + 1/13106661870132*a^28 - 1/4368887290044*a^26 + 9257801039/9829996402599*a^24 - 29349668189/13106661870132*a^22 + 11712482075/4368887290044*a^20 + 2117068222525/39319985610396*a^18 - 17551756493/161810640372*a^16 + 89087782901/1092221822511*a^14 - 5001641332921/39319985610396*a^12 - 706711017953/13106661870132*a^10 - 1981734423185/4368887290044*a^8 - 15464536204685/39319985610396*a^6 + 324155339690/1092221822511*a^4 + 73440071909/485431921116*a^2 - 5866965635/13484220031, 1/117959956831188*a^31 + 1/39319985610396*a^29 - 1/13106661870132*a^27 + 9257801039/29489989207797*a^25 + 46004156045/19659992805198*a^23 + 11712482075/13106661870132*a^21 + 660772459177/117959956831188*a^19 - 535523969305/4368887290044*a^17 - 32270197378/3276665467533*a^15 + 2414177534839/58979978415594*a^13 + 1235016666511/39319985610396*a^11 + 121803901651/13106661870132*a^9 - 19833423494729/117959956831188*a^7 + 1133208541550/3276665467533*a^5 + 191788566311/728147881674*a^3 + 7617254396/40452660093*a

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)

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:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{1620997322}{29489989207797} a^{31} - \frac{6359036717}{39319985610396} a^{29} + \frac{1620997322}{3276665467533} a^{27} + \frac{137784772370}{29489989207797} a^{25} + \frac{61597898236}{9829996402599} a^{23} + \frac{273948547418}{3276665467533} a^{21} + \frac{12532743803623}{117959956831188} a^{19} - \frac{1340564785294}{1092221822511} a^{17} - \frac{20121439757986}{3276665467533} a^{15} + \frac{72494242234484}{29489989207797} a^{13} + \frac{22280608190890}{9829996402599} a^{11} + \frac{66474293300125}{13106661870132} a^{9} - \frac{26018628015422}{29489989207797} a^{7} - \frac{2390971049950}{3276665467533} a^{5} + \frac{236665609012}{364073940837} a^{3} + \frac{6483989288}{40452660093} a $ -(1620997322)/(29489989207797)a^(31) - (6359036717)/(39319985610396)a^(29) + (1620997322)/(3276665467533)a^(27) + (137784772370)/(29489989207797)a^(25) + (61597898236)/(9829996402599)a^(23) + (273948547418)/(3276665467533)a^(21) + (12532743803623)/(117959956831188)a^(19) - (1340564785294)/(1092221822511)a^(17) - (20121439757986)/(3276665467533)a^(15) + (72494242234484)/(29489989207797)a^(13) + (22280608190890)/(9829996402599)a^(11) + (66474293300125)/(13106661870132)a^(9) - (26018628015422)/(29489989207797)a^(7) - (2390971049950)/(3276665467533)a^(5) + (236665609012)/(364073940837)a^(3) + (6483989288)/(40452660093)a (order $20$)	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:	$\frac{58743065}{485431921116}a^{30}+\frac{1008990349}{4368887290044}a^{28}-\frac{1071618067}{728147881674}a^{26}-\frac{1478053829}{161810640372}a^{24}-\frac{3188866615}{1092221822511}a^{22}-\frac{246802642151}{1456295763348}a^{20}-\frac{22841210357}{485431921116}a^{18}+\frac{6479261533811}{2184443645022}a^{16}+\frac{1725240793109}{161810640372}a^{14}-\frac{797163534302}{40452660093}a^{12}+\frac{3199122409961}{4368887290044}a^{10}+\frac{468453620623}{1456295763348}a^{8}+\frac{31309304773}{80905320186}a^{6}-\frac{1954165868063}{4368887290044}a^{4}-\frac{379804766935}{121357980279}a^{2}+\frac{2161056591}{13484220031}$, $\frac{4551235}{161810640372}a^{31}-\frac{4134277}{161810640372}a^{30}-\frac{4551235}{53936880124}a^{29}+\frac{4134277}{53936880124}a^{28}-\frac{386854975}{485431921116}a^{27}+\frac{351413545}{485431921116}a^{26}-\frac{86473465}{80905320186}a^{25}+\frac{78551263}{80905320186}a^{24}+\frac{8173015357}{728147881674}a^{23}-\frac{549928801}{53936880124}a^{22}-\frac{4756040575}{242715960558}a^{21}+\frac{4320319465}{242715960558}a^{20}+\frac{11291614035}{53936880124}a^{19}-\frac{10257141237}{53936880124}a^{18}+\frac{56494480055}{53936880124}a^{17}-\frac{51318780401}{53936880124}a^{16}-\frac{101770165835}{242715960558}a^{15}+\frac{92446567997}{242715960558}a^{14}-\frac{15128911721125}{728147881674}a^{13}+\frac{3053780886151}{161810640372}a^{12}+\frac{8460745865}{26968440062}a^{11}-\frac{7685620943}{26968440062}a^{10}+\frac{73051872985}{485431921116}a^{9}-\frac{66359280127}{485431921116}a^{8}+\frac{6713071625}{53936880124}a^{7}-\frac{6098058575}{53936880124}a^{6}-\frac{2990161395}{26968440062}a^{5}+\frac{2716219989}{26968440062}a^{4}-\frac{1642097132039}{728147881674}a^{3}+\frac{1479806698723}{485431921116}a^{2}+\frac{3317850315}{53936880124}a-\frac{3013887933}{53936880124}$, $\frac{4551235}{161810640372}a^{31}-\frac{4551235}{53936880124}a^{29}-\frac{386854975}{485431921116}a^{27}-\frac{86473465}{80905320186}a^{25}+\frac{8173015357}{728147881674}a^{23}-\frac{4756040575}{242715960558}a^{21}+\frac{11291614035}{53936880124}a^{19}+\frac{56494480055}{53936880124}a^{17}-\frac{101770165835}{242715960558}a^{15}-\frac{15128911721125}{728147881674}a^{13}+\frac{8460745865}{26968440062}a^{11}+\frac{73051872985}{485431921116}a^{9}+\frac{6713071625}{53936880124}a^{7}-\frac{2990161395}{26968440062}a^{5}-\frac{1642097132039}{728147881674}a^{3}+\frac{3317850315}{53936880124}a-1$, $\frac{382715555}{728147881674}a^{31}-\frac{2547203}{242715960558}a^{30}+\frac{11098751095}{6553330935066}a^{29}-\frac{17987631085}{4368887290044}a^{27}-\frac{32724725015}{728147881674}a^{25}-\frac{946455567515}{13106661870132}a^{23}-\frac{3648427385815}{4368887290044}a^{21}+\frac{13848020425}{485431921116}a^{20}-\frac{951048154175}{728147881674}a^{19}+\frac{145383306024515}{13106661870132}a^{17}+\frac{14753941188815}{242715960558}a^{15}-\frac{2227787245655}{485431921116}a^{13}+\frac{32276316330925}{13106661870132}a^{11}-\frac{11080416333985}{485431921116}a^{10}+\frac{2224725521215}{2184443645022}a^{9}+\frac{732134856715}{485431921116}a^{7}+\frac{66197036516161}{6553330935066}a^{5}-\frac{24111079965}{53936880124}a^{3}+\frac{30999959955}{53936880124}a-\frac{118182362941}{53936880124}$, $\frac{83548727}{728147881674}a^{31}+\frac{4627873}{1456295763348}a^{30}+\frac{2422913083}{6553330935066}a^{29}-\frac{3926790169}{4368887290044}a^{27}-\frac{529221843}{53936880124}a^{25}-\frac{206616001871}{13106661870132}a^{23}-\frac{796470014491}{4368887290044}a^{21}-\frac{12577701337}{1456295763348}a^{20}-\frac{207618586595}{728147881674}a^{19}+\frac{31737905571671}{13106661870132}a^{17}+\frac{19327894281113}{1456295763348}a^{15}-\frac{486337139867}{485431921116}a^{13}+\frac{7046081891545}{13106661870132}a^{11}+\frac{10064640608101}{1456295763348}a^{10}+\frac{485668750051}{2184443645022}a^{9}+\frac{159828714751}{485431921116}a^{7}+\frac{8969900914163}{13106661870132}a^{5}-\frac{5263569801}{53936880124}a^{3}+\frac{6767446887}{53936880124}a+\frac{18347211997}{13484220031}$, $\frac{5385151}{485431921116}a^{31}-\frac{633743243}{3276665467533}a^{30}-\frac{5385151}{161810640372}a^{29}-\frac{2521062209}{4368887290044}a^{28}-\frac{457737835}{1456295763348}a^{27}+\frac{633743243}{364073940837}a^{26}-\frac{102317869}{242715960558}a^{25}+\frac{53868175655}{3276665467533}a^{24}+\frac{1611828074}{364073940837}a^{23}+\frac{24082243234}{1092221822511}a^{22}-\frac{5627482795}{728147881674}a^{21}+\frac{107102608067}{364073940837}a^{20}+\frac{4453519877}{53936880124}a^{19}+\frac{5184738020971}{13106661870132}a^{18}+\frac{66845879363}{161810640372}a^{17}-\frac{524105661961}{121357980279}a^{16}-\frac{120417361511}{728147881674}a^{15}-\frac{7866654875359}{364073940837}a^{14}-\frac{2983784531336}{364073940837}a^{13}+\frac{28342265313446}{3276665467533}a^{12}+\frac{10010995709}{80905320186}a^{11}+\frac{8710800875035}{1092221822511}a^{10}+\frac{86437058701}{1456295763348}a^{9}+\frac{639606016133}{1456295763348}a^{8}+\frac{7943097725}{161810640372}a^{7}-\frac{10172212793393}{3276665467533}a^{6}-\frac{1179348069}{26968440062}a^{5}-\frac{934771283425}{364073940837}a^{4}+\frac{40212763621}{364073940837}a^{3}+\frac{92526513478}{40452660093}a^{2}+\frac{1308591693}{53936880124}a+\frac{7604918916}{13484220031}$, $\frac{1067370893}{19659992805198}a^{30}+\frac{2669837713}{13106661870132}a^{28}-\frac{340821058}{1092221822511}a^{26}-\frac{191457485497}{39319985610396}a^{24}-\frac{135776310179}{13106661870132}a^{22}-\frac{400413910603}{4368887290044}a^{20}-\frac{6967628512007}{39319985610396}a^{18}+\frac{128193421138}{121357980279}a^{16}+\frac{30374628307637}{4368887290044}a^{14}+\frac{139743045531569}{39319985610396}a^{12}+\frac{14752881487417}{13106661870132}a^{10}-\frac{39510630174509}{4368887290044}a^{8}-\frac{64242428291030}{9829996402599}a^{6}-\frac{2303574971737}{4368887290044}a^{4}+\frac{343811761811}{485431921116}a^{2}+\frac{51504171059}{53936880124}$, $\frac{484544159}{19659992805198}a^{30}+\frac{1007819963}{6553330935066}a^{28}+\frac{46505846}{1092221822511}a^{26}-\frac{107654257837}{39319985610396}a^{24}-\frac{128781580613}{13106661870132}a^{22}-\frac{213027379765}{4368887290044}a^{20}-\frac{3457675446811}{19659992805198}a^{18}+\frac{126134769830}{364073940837}a^{16}+\frac{19596281372129}{4368887290044}a^{14}+\frac{329751503020955}{39319985610396}a^{12}-\frac{22712166379157}{13106661870132}a^{10}-\frac{7884428296321}{2184443645022}a^{8}+\frac{5452042100299}{9829996402599}a^{6}+\frac{2458264701323}{4368887290044}a^{4}+\frac{944571224149}{485431921116}a^{2}-\frac{23971782707}{53936880124}$, $\frac{8462161409}{58979978415594}a^{31}-\frac{502306511}{9829996402599}a^{30}+\frac{5025908377}{9829996402599}a^{29}-\frac{1528002677}{13106661870132}a^{28}-\frac{6422376581}{6553330935066}a^{27}+\frac{618546668}{1092221822511}a^{26}-\frac{1504881530977}{117959956831188}a^{25}+\frac{155815457885}{39319985610396}a^{24}-\frac{235805162678}{9829996402599}a^{23}+\frac{8651728306}{3276665467533}a^{22}-\frac{3053823287683}{13106661870132}a^{21}+\frac{324434757149}{4368887290044}a^{20}-\frac{12596130537692}{29489989207797}a^{19}+\frac{2143968189439}{39319985610396}a^{18}+\frac{19244729101721}{6553330935066}a^{17}-\frac{440268502715}{364073940837}a^{16}+\frac{232725358305965}{13106661870132}a^{15}-\frac{21074091934345}{4368887290044}a^{14}+\frac{125881794566129}{29489989207797}a^{13}+\frac{62476001525057}{9829996402599}a^{12}-\frac{105688907703239}{39319985610396}a^{11}-\frac{11280577915247}{13106661870132}a^{10}-\frac{21318046921610}{3276665467533}a^{9}-\frac{30550164552551}{4368887290044}a^{8}+\frac{78937682132837}{58979978415594}a^{7}+\frac{44058492480490}{9829996402599}a^{6}+\frac{878681010139}{161810640372}a^{5}+\frac{1731046265237}{4368887290044}a^{4}-\frac{124740822253}{121357980279}a^{3}-\frac{100512055570}{121357980279}a^{2}-\frac{184846546681}{161810640372}a+\frac{16842041985}{53936880124}$, $\frac{3380355103}{19659992805198}a^{31}+\frac{82786925}{728147881674}a^{30}+\frac{6939964313}{13106661870132}a^{29}+\frac{1008990349}{4368887290044}a^{28}-\frac{6994184417}{4368887290044}a^{27}-\frac{1071618067}{728147881674}a^{26}-\frac{594505675445}{39319985610396}a^{25}-\frac{1478053829}{161810640372}a^{24}-\frac{132889503923}{6553330935066}a^{23}-\frac{3188866615}{1092221822511}a^{22}-\frac{277875240218}{1092221822511}a^{21}-\frac{24204031357}{161810640372}a^{20}-\frac{14175507827779}{39319985610396}a^{19}-\frac{22841210357}{485431921116}a^{18}+\frac{5784190512859}{1456295763348}a^{17}+\frac{6479261533811}{2184443645022}a^{16}+\frac{86818811168221}{4368887290044}a^{15}+\frac{1725240793109}{161810640372}a^{14}-\frac{156396957748537}{19659992805198}a^{13}-\frac{797163534302}{40452660093}a^{12}-\frac{66347157979255}{3276665467533}a^{11}-\frac{66330702771601}{4368887290044}a^{10}-\frac{13331530776925}{4368887290044}a^{9}+\frac{468453620623}{1456295763348}a^{8}+\frac{112263654077267}{39319985610396}a^{7}+\frac{31309304773}{80905320186}a^{6}+\frac{10316422015075}{4368887290044}a^{5}-\frac{1954165868063}{4368887290044}a^{4}-\frac{510575462441}{242715960558}a^{3}-\frac{379804766935}{121357980279}a^{2}-\frac{212523749365}{80905320186}a-\frac{90086168713}{53936880124}$, $\frac{108890209}{1456295763348}a^{31}+\frac{23173133}{161810640372}a^{30}+\frac{2253110693}{13106661870132}a^{29}+\frac{301180123}{485431921116}a^{28}-\frac{3552480271}{4368887290044}a^{27}-\frac{145155295}{242715960558}a^{26}-\frac{2135460889}{364073940837}a^{25}-\frac{9800837545}{728147881674}a^{24}-\frac{28320357565}{6553330935066}a^{23}-\frac{4051140841}{121357980279}a^{22}-\frac{118946277811}{1092221822511}a^{21}-\frac{121520130835}{485431921116}a^{20}-\frac{110200831981}{1456295763348}a^{19}-\frac{32876779187}{53936880124}a^{18}+\frac{23028231211397}{13106661870132}a^{17}+\frac{633511134773}{242715960558}a^{16}+\frac{2620028583425}{364073940837}a^{15}+\frac{14490319932601}{728147881674}a^{14}-\frac{6372425823679}{728147881674}a^{13}+\frac{703836558934}{40452660093}a^{12}+\frac{1383043532122}{3276665467533}a^{11}+\frac{245487226751}{485431921116}a^{10}+\frac{798378312731}{4368887290044}a^{9}+\frac{92378694203}{485431921116}a^{8}+\frac{112530087289}{485431921116}a^{7}+\frac{10010739545}{26968440062}a^{6}+\frac{11013731477267}{3276665467533}a^{5}+\frac{1473231998747}{728147881674}a^{4}+\frac{40990072253}{728147881674}a^{3}+\frac{352532960527}{121357980279}a^{2}+\frac{5064349311}{53936880124}a+\frac{1734915753}{13484220031}$, $\frac{7021507313}{117959956831188}a^{31}+\frac{52494727}{1456295763348}a^{30}+\frac{6359036717}{39319985610396}a^{29}-\frac{4134277}{53936880124}a^{28}-\frac{1620997322}{3276665467533}a^{27}-\frac{351413545}{485431921116}a^{26}-\frac{137784772370}{29489989207797}a^{25}-\frac{78551263}{80905320186}a^{24}-\frac{61597898236}{9829996402599}a^{23}+\frac{549928801}{53936880124}a^{22}-\frac{314535531086}{3276665467533}a^{21}-\frac{67471762453}{1456295763348}a^{20}-\frac{12532743803623}{117959956831188}a^{19}+\frac{10257141237}{53936880124}a^{18}+\frac{1340564785294}{1092221822511}a^{17}+\frac{51318780401}{53936880124}a^{16}+\frac{20121439757986}{3276665467533}a^{15}-\frac{92446567997}{242715960558}a^{14}-\frac{72494242234484}{29489989207797}a^{13}-\frac{3053780886151}{161810640372}a^{12}+\frac{75157647243290}{9829996402599}a^{11}+\frac{33656270838853}{1456295763348}a^{10}-\frac{66474293300125}{13106661870132}a^{9}+\frac{66359280127}{485431921116}a^{8}+\frac{26018628015422}{29489989207797}a^{7}+\frac{6098058575}{53936880124}a^{6}+\frac{2390971049950}{3276665467533}a^{5}-\frac{2716219989}{26968440062}a^{4}-\frac{236665609012}{364073940837}a^{3}-\frac{1479806698723}{485431921116}a^{2}-\frac{174316794427}{161810640372}a+\frac{122700389291}{26968440062}$, $\frac{1902341758}{9829996402599}a^{30}+\frac{8977875463}{13106661870132}a^{28}-\frac{3019598669}{2184443645022}a^{26}-\frac{680601546127}{39319985610396}a^{24}-\frac{206670997183}{6553330935066}a^{22}-\frac{676583567219}{2184443645022}a^{20}-\frac{22162648838261}{39319985610396}a^{18}+\frac{8826490841975}{2184443645022}a^{16}+\frac{104631416957783}{4368887290044}a^{14}+\frac{82249140559699}{19659992805198}a^{12}-\frac{57006098881987}{6553330935066}a^{10}-\frac{28444279392971}{4368887290044}a^{8}+\frac{102150755093975}{19659992805198}a^{6}+\frac{4071695136415}{1456295763348}a^{4}-\frac{252376118945}{242715960558}a^{2}-\frac{50850631561}{26968440062}$, $\frac{811177423}{58979978415594}a^{31}-\frac{3344267387}{19659992805198}a^{29}-\frac{4455100879}{6553330935066}a^{27}+\frac{26287552795}{29489989207797}a^{25}+\frac{305111832223}{19659992805198}a^{23}-\frac{17330577043}{6553330935066}a^{21}+\frac{17091606766981}{58979978415594}a^{19}+\frac{1386524706511}{2184443645022}a^{17}-\frac{10598420684522}{3276665467533}a^{15}-\frac{13\!\cdots\!05}{58979978415594}a^{13}+\frac{311346591636865}{19659992805198}a^{11}-\frac{10742842427717}{6553330935066}a^{9}-\frac{90638482784777}{58979978415594}a^{7}-\frac{6955260652787}{3276665467533}a^{5}-\frac{380002635023}{242715960558}a^{3}+\frac{158926684154}{40452660093}a$, $\frac{10335015053}{39319985610396}a^{30}+\frac{2877331748}{3276665467533}a^{28}-\frac{8568215459}{4368887290044}a^{26}-\frac{223765468469}{9829996402599}a^{24}-\frac{127299066946}{3276665467533}a^{22}-\frac{1840783012661}{4368887290044}a^{20}-\frac{6876416695918}{9829996402599}a^{18}+\frac{23929192163527}{4368887290044}a^{16}+\frac{33988771095229}{1092221822511}a^{14}+\frac{12894100899073}{9829996402599}a^{12}-\frac{1690559442865}{13106661870132}a^{10}-\frac{3210773294368}{1092221822511}a^{8}+\frac{120654893225345}{39319985610396}a^{6}+\frac{2259455920457}{1092221822511}a^{4}+\frac{26005003154}{121357980279}a^{2}+\frac{16439982312}{13484220031}$ 58743065/485431921116a^30 + 1008990349/4368887290044a^28 - 1071618067/728147881674a^26 - 1478053829/161810640372a^24 - 3188866615/1092221822511a^22 - 246802642151/1456295763348a^20 - 22841210357/485431921116a^18 + 6479261533811/2184443645022a^16 + 1725240793109/161810640372a^14 - 797163534302/40452660093a^12 + 3199122409961/4368887290044a^10 + 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400413910603/4368887290044a^20 - 6967628512007/39319985610396a^18 + 128193421138/121357980279a^16 + 30374628307637/4368887290044a^14 + 139743045531569/39319985610396a^12 + 14752881487417/13106661870132a^10 - 39510630174509/4368887290044a^8 - 64242428291030/9829996402599a^6 - 2303574971737/4368887290044a^4 + 343811761811/485431921116a^2 + 51504171059/53936880124, 484544159/19659992805198a^30 + 1007819963/6553330935066a^28 + 46505846/1092221822511a^26 - 107654257837/39319985610396a^24 - 128781580613/13106661870132a^22 - 213027379765/4368887290044a^20 - 3457675446811/19659992805198a^18 + 126134769830/364073940837a^16 + 19596281372129/4368887290044a^14 + 329751503020955/39319985610396a^12 - 22712166379157/13106661870132a^10 - 7884428296321/2184443645022a^8 + 5452042100299/9829996402599a^6 + 2458264701323/4368887290044a^4 + 944571224149/485431921116a^2 - 23971782707/53936880124, 8462161409/58979978415594a^31 - 502306511/9829996402599a^30 + 5025908377/9829996402599a^29 - 1528002677/13106661870132a^28 - 6422376581/6553330935066a^27 + 618546668/1092221822511a^26 - 1504881530977/117959956831188a^25 + 155815457885/39319985610396a^24 - 235805162678/9829996402599a^23 + 8651728306/3276665467533a^22 - 3053823287683/13106661870132a^21 + 324434757149/4368887290044a^20 - 12596130537692/29489989207797a^19 + 2143968189439/39319985610396a^18 + 19244729101721/6553330935066a^17 - 440268502715/364073940837a^16 + 232725358305965/13106661870132a^15 - 21074091934345/4368887290044a^14 + 125881794566129/29489989207797a^13 + 62476001525057/9829996402599a^12 - 105688907703239/39319985610396a^11 - 11280577915247/13106661870132a^10 - 21318046921610/3276665467533a^9 - 30550164552551/4368887290044a^8 + 78937682132837/58979978415594a^7 + 44058492480490/9829996402599a^6 + 878681010139/161810640372a^5 + 1731046265237/4368887290044a^4 - 124740822253/121357980279a^3 - 100512055570/121357980279a^2 - 184846546681/161810640372a + 16842041985/53936880124, 3380355103/19659992805198a^31 + 82786925/728147881674a^30 + 6939964313/13106661870132a^29 + 1008990349/4368887290044a^28 - 6994184417/4368887290044a^27 - 1071618067/728147881674a^26 - 594505675445/39319985610396a^25 - 1478053829/161810640372a^24 - 132889503923/6553330935066a^23 - 3188866615/1092221822511a^22 - 277875240218/1092221822511a^21 - 24204031357/161810640372a^20 - 14175507827779/39319985610396a^19 - 22841210357/485431921116a^18 + 5784190512859/1456295763348a^17 + 6479261533811/2184443645022a^16 + 86818811168221/4368887290044a^15 + 1725240793109/161810640372a^14 - 156396957748537/19659992805198a^13 - 797163534302/40452660093a^12 - 66347157979255/3276665467533a^11 - 66330702771601/4368887290044a^10 - 13331530776925/4368887290044a^9 + 468453620623/1456295763348a^8 + 112263654077267/39319985610396a^7 + 31309304773/80905320186a^6 + 10316422015075/4368887290044a^5 - 1954165868063/4368887290044a^4 - 510575462441/242715960558a^3 - 379804766935/121357980279a^2 - 212523749365/80905320186a - 90086168713/53936880124, 108890209/1456295763348a^31 + 23173133/161810640372a^30 + 2253110693/13106661870132a^29 + 301180123/485431921116a^28 - 3552480271/4368887290044a^27 - 145155295/242715960558a^26 - 2135460889/364073940837a^25 - 9800837545/728147881674a^24 - 28320357565/6553330935066a^23 - 4051140841/121357980279a^22 - 118946277811/1092221822511a^21 - 121520130835/485431921116a^20 - 110200831981/1456295763348a^19 - 32876779187/53936880124a^18 + 23028231211397/13106661870132a^17 + 633511134773/242715960558a^16 + 2620028583425/364073940837a^15 + 14490319932601/728147881674a^14 - 6372425823679/728147881674a^13 + 703836558934/40452660093a^12 + 1383043532122/3276665467533a^11 + 245487226751/485431921116a^10 + 798378312731/4368887290044a^9 + 92378694203/485431921116a^8 + 112530087289/485431921116a^7 + 10010739545/26968440062a^6 + 11013731477267/3276665467533a^5 + 1473231998747/728147881674a^4 + 40990072253/728147881674a^3 + 352532960527/121357980279a^2 + 5064349311/53936880124a + 1734915753/13484220031, 7021507313/117959956831188a^31 + 52494727/1456295763348a^30 + 6359036717/39319985610396a^29 - 4134277/53936880124a^28 - 1620997322/3276665467533a^27 - 351413545/485431921116a^26 - 137784772370/29489989207797a^25 - 78551263/80905320186a^24 - 61597898236/9829996402599a^23 + 549928801/53936880124a^22 - 314535531086/3276665467533a^21 - 67471762453/1456295763348a^20 - 12532743803623/117959956831188a^19 + 10257141237/53936880124a^18 + 1340564785294/1092221822511a^17 + 51318780401/53936880124a^16 + 20121439757986/3276665467533a^15 - 92446567997/242715960558a^14 - 72494242234484/29489989207797a^13 - 3053780886151/161810640372a^12 + 75157647243290/9829996402599a^11 + 33656270838853/1456295763348a^10 - 66474293300125/13106661870132a^9 + 66359280127/485431921116a^8 + 26018628015422/29489989207797a^7 + 6098058575/53936880124a^6 + 2390971049950/3276665467533a^5 - 2716219989/26968440062a^4 - 236665609012/364073940837a^3 - 1479806698723/485431921116a^2 - 174316794427/161810640372a + 122700389291/26968440062, 1902341758/9829996402599a^30 + 8977875463/13106661870132a^28 - 3019598669/2184443645022a^26 - 680601546127/39319985610396a^24 - 206670997183/6553330935066a^22 - 676583567219/2184443645022a^20 - 22162648838261/39319985610396a^18 + 8826490841975/2184443645022a^16 + 104631416957783/4368887290044a^14 + 82249140559699/19659992805198a^12 - 57006098881987/6553330935066a^10 - 28444279392971/4368887290044a^8 + 102150755093975/19659992805198a^6 + 4071695136415/1456295763348a^4 - 252376118945/242715960558a^2 - 50850631561/26968440062, 811177423/58979978415594a^31 - 3344267387/19659992805198a^29 - 4455100879/6553330935066a^27 + 26287552795/29489989207797a^25 + 305111832223/19659992805198a^23 - 17330577043/6553330935066a^21 + 17091606766981/58979978415594a^19 + 1386524706511/2184443645022a^17 - 10598420684522/3276665467533a^15 - 1321435138312105/58979978415594a^13 + 311346591636865/19659992805198a^11 - 10742842427717/6553330935066a^9 - 90638482784777/58979978415594a^7 - 6955260652787/3276665467533a^5 - 380002635023/242715960558a^3 + 158926684154/40452660093a, 10335015053/39319985610396a^30 + 2877331748/3276665467533a^28 - 8568215459/4368887290044a^26 - 223765468469/9829996402599a^24 - 127299066946/3276665467533a^22 - 1840783012661/4368887290044a^20 - 6876416695918/9829996402599a^18 + 23929192163527/4368887290044a^16 + 33988771095229/1092221822511a^14 + 12894100899073/9829996402599a^12 - 1690559442865/13106661870132a^10 - 3210773294368/1092221822511a^8 + 120654893225345/39319985610396a^6 + 2259455920457/1092221822511a^4 + 26005003154/121357980279*a^2 + 16439982312/13484220031 (assuming GRH)	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:	$ 4622596654413.537 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 4622596654413.537 \cdot 64}{20\cdot\sqrt{138956997215838851269528412416000000000000000000000000}}\cr\approx \mathstrut & 0.234138947805810 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561)

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^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561, 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^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561);

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^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561);

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

$C_2\times C_4^2$ (as 32T36):

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 32

The 32 conjugacy class representatives for $C_2\times C_4^2$

Character table for $C_2\times C_4^2$

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{65}) $, $\Q(\sqrt{-65}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{13}) $, $\Q(\sqrt{-13}) $, $\Q(i, \sqrt{65})$, 4.0.4394000.2, 4.4.274625.1, $\Q(i, \sqrt{5})$, $\Q(i, \sqrt{13})$, $\Q(\sqrt{5}, \sqrt{13})$, $\Q(\sqrt{-5}, \sqrt{-13})$, 4.0.4394000.1, 4.4.274625.2, $\Q(\sqrt{5}, \sqrt{-13})$, $\Q(\sqrt{-5}, \sqrt{13})$, 4.0.21125.1, 4.4.338000.1, $\Q(\zeta_{5})$, $\Q(\zeta_{20})^+$, 4.4.35152.1, 4.0.2197.1, 4.4.878800.1, 4.0.54925.1, 8.0.19307236000000.1, 8.0.4569760000.1, 8.0.19307236000000.4, 8.0.19307236000000.5, 8.0.19307236000000.3, 8.8.75418890625.1, 8.0.19307236000000.2, 8.0.114244000000.2, $\Q(\zeta_{20})$, 8.0.1235663104.1, 8.0.772289440000.3, 8.0.446265625.1, 8.8.114244000000.1, 8.8.772289440000.1, 8.0.3016755625.1, 8.0.114244000000.1, 8.0.114244000000.3, 8.0.772289440000.1, 8.0.772289440000.2, 16.0.372769361959696000000000000.1, 16.0.13051691536000000000000.1, 16.0.596430979135513600000000.1, 16.0.372769361959696000000000000.3, 16.0.372769361959696000000000000.2, 16.0.5688009063105712890625.1, 16.16.372769361959696000000000000.2

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

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

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

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

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

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.4.0.1}{4} }^{8}$	R	${\href{/padicField/7.4.0.1}{4} }^{8}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	R	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.4.0.1}{4} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	${\href{/padicField/29.2.0.1}{2} }^{16}$	${\href{/padicField/31.4.0.1}{4} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{8}$

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
$2$ 2	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$5$ 5	5.16.12.1	$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$
$5$ 5	5.16.12.1	$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$
$13$ 13	13.16.12.1	$x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$
$13$ 13	13.16.12.1	$x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$

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