LMFDB - Number field 32.0.111539154030155436264757885742310633531656776974336.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536)

gp: K = bnfinit(y^32 + 7*y^30 + 23*y^28 + 38*y^26 + 11*y^24 - 94*y^22 - 186*y^20 - 24*y^18 + 385*y^16 - 96*y^14 - 2976*y^12 - 6016*y^10 + 2816*y^8 + 38912*y^6 + 94208*y^4 + 114688*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536)

$ x^{32} + 7 x^{30} + 23 x^{28} + 38 x^{26} + 11 x^{24} - 94 x^{22} - 186 x^{20} - 24 x^{18} + \cdots + 65536 $ x^32 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536

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:	$111539154030155436264757885742310633531656776974336$ 111539154030155436264757885742310633531656776974336 $\medspace = 2^{32}\cdot 3^{16}\cdot 89^{4}\cdot 1327^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$36.64$	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:	$2\cdot 3^{1/2}89^{1/2}1327^{1/2}\approx 1190.4772152376543$
Ramified primes:	$2$, $3$, $89$, $1327$ 2, 3, 89, 1327	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{15}-\frac{1}{6}a^{13}-\frac{1}{6}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{1}{12}a^{16}-\frac{1}{12}a^{14}-\frac{1}{6}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{5}{12}a^{2}+\frac{1}{3}$, $\frac{1}{24}a^{19}-\frac{1}{24}a^{17}-\frac{1}{24}a^{15}-\frac{1}{12}a^{13}+\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{12}a^{7}+\frac{1}{3}a^{5}-\frac{7}{24}a^{3}-\frac{1}{3}a$, $\frac{1}{48}a^{20}-\frac{1}{48}a^{18}-\frac{1}{48}a^{16}-\frac{1}{24}a^{14}-\frac{5}{48}a^{12}+\frac{5}{24}a^{10}+\frac{11}{24}a^{8}-\frac{1}{6}a^{6}+\frac{17}{48}a^{4}+\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{96}a^{21}-\frac{1}{96}a^{19}-\frac{1}{96}a^{17}+\frac{7}{48}a^{15}-\frac{5}{96}a^{13}+\frac{7}{16}a^{11}-\frac{7}{16}a^{9}-\frac{5}{12}a^{7}+\frac{11}{32}a^{5}-\frac{5}{12}a^{3}$, $\frac{1}{192}a^{22}-\frac{1}{192}a^{20}-\frac{1}{192}a^{18}+\frac{7}{96}a^{16}-\frac{5}{192}a^{14}+\frac{5}{96}a^{12}+\frac{11}{96}a^{10}+\frac{7}{24}a^{8}+\frac{65}{192}a^{6}-\frac{5}{24}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{384}a^{23}-\frac{1}{384}a^{21}-\frac{1}{384}a^{19}+\frac{7}{192}a^{17}-\frac{5}{384}a^{15}-\frac{9}{64}a^{13}+\frac{25}{64}a^{11}+\frac{7}{48}a^{9}+\frac{43}{128}a^{7}+\frac{19}{48}a^{5}-\frac{1}{2}a$, $\frac{1}{2304}a^{24}-\frac{1}{2304}a^{22}-\frac{17}{2304}a^{20}+\frac{47}{1152}a^{18}-\frac{53}{2304}a^{16}-\frac{107}{1152}a^{14}+\frac{115}{1152}a^{12}+\frac{139}{288}a^{10}+\frac{673}{2304}a^{8}-\frac{61}{288}a^{6}+\frac{7}{144}a^{4}+\frac{2}{9}a^{2}-\frac{2}{9}$, $\frac{1}{4608}a^{25}-\frac{1}{4608}a^{23}-\frac{17}{4608}a^{21}+\frac{47}{2304}a^{19}-\frac{53}{4608}a^{17}-\frac{107}{2304}a^{15}-\frac{269}{2304}a^{13}+\frac{43}{576}a^{11}-\frac{1631}{4608}a^{9}+\frac{35}{576}a^{7}-\frac{137}{288}a^{5}+\frac{4}{9}a^{3}-\frac{5}{18}a$, $\frac{1}{9216}a^{26}-\frac{1}{9216}a^{24}-\frac{17}{9216}a^{22}+\frac{47}{4608}a^{20}-\frac{53}{9216}a^{18}+\frac{661}{4608}a^{16}-\frac{269}{4608}a^{14}+\frac{43}{1152}a^{12}+\frac{2977}{9216}a^{10}-\frac{349}{1152}a^{8}-\frac{137}{576}a^{6}+\frac{2}{9}a^{4}+\frac{13}{36}a^{2}-\frac{1}{3}$, $\frac{1}{18432}a^{27}-\frac{1}{18432}a^{25}-\frac{17}{18432}a^{23}+\frac{47}{9216}a^{21}-\frac{53}{18432}a^{19}+\frac{661}{9216}a^{17}-\frac{269}{9216}a^{15}+\frac{43}{2304}a^{13}+\frac{2977}{18432}a^{11}+\frac{803}{2304}a^{9}+\frac{439}{1152}a^{7}+\frac{1}{9}a^{5}-\frac{23}{72}a^{3}-\frac{1}{6}a$, $\frac{1}{6746112}a^{28}+\frac{121}{2248704}a^{26}-\frac{317}{6746112}a^{24}+\frac{3257}{3373056}a^{22}+\frac{3803}{749568}a^{20}-\frac{76057}{3373056}a^{18}+\frac{189871}{3373056}a^{16}+\frac{1193}{70272}a^{14}+\frac{89921}{6746112}a^{12}+\frac{527425}{1686528}a^{10}+\frac{6311}{15616}a^{8}-\frac{22805}{105408}a^{6}+\frac{2575}{26352}a^{4}-\frac{431}{2196}a^{2}+\frac{553}{1647}$, $\frac{1}{13492224}a^{29}+\frac{121}{4497408}a^{27}-\frac{317}{13492224}a^{25}+\frac{3257}{6746112}a^{23}+\frac{3803}{1499136}a^{21}-\frac{76057}{6746112}a^{19}+\frac{189871}{6746112}a^{17}+\frac{1193}{140544}a^{15}-\frac{2158783}{13492224}a^{13}+\frac{1651777}{3373056}a^{11}-\frac{9305}{31232}a^{9}-\frac{93077}{210816}a^{7}+\frac{2575}{52704}a^{5}+\frac{1033}{4392}a^{3}-\frac{1643}{3294}a$, $\frac{1}{1969864704}a^{30}-\frac{137}{1969864704}a^{28}+\frac{64135}{1969864704}a^{26}+\frac{18091}{984932352}a^{24}-\frac{2180405}{1969864704}a^{22}+\frac{5726105}{984932352}a^{20}+\frac{13104481}{328310784}a^{18}+\frac{4764025}{246233088}a^{16}+\frac{220845377}{1969864704}a^{14}-\frac{1613989}{41038848}a^{12}-\frac{477989}{30779136}a^{10}-\frac{447275}{1923696}a^{8}-\frac{219485}{3847392}a^{6}-\frac{554663}{1923696}a^{4}+\frac{21421}{120231}a^{2}+\frac{34819}{120231}$, $\frac{1}{3939729408}a^{31}-\frac{137}{3939729408}a^{29}+\frac{64135}{3939729408}a^{27}+\frac{18091}{1969864704}a^{25}-\frac{2180405}{3939729408}a^{23}+\frac{5726105}{1969864704}a^{21}+\frac{13104481}{656621568}a^{19}+\frac{4764025}{492466176}a^{17}+\frac{220845377}{3939729408}a^{15}+\frac{12065627}{82077696}a^{13}-\frac{20997413}{61558272}a^{11}+\frac{1476421}{3847392}a^{9}+\frac{2345443}{7694784}a^{7}+\frac{1369033}{3847392}a^{5}-\frac{58733}{240462}a^{3}+\frac{37448}{120231}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/3*a^12 + 1/3*a^10 - 1/3*a^6 + 1/3*a^2 + 1/3, 1/3*a^13 + 1/3*a^11 - 1/3*a^7 + 1/3*a^3 + 1/3*a, 1/3*a^14 - 1/3*a^10 - 1/3*a^8 + 1/3*a^6 + 1/3*a^4 - 1/3, 1/3*a^15 - 1/3*a^11 - 1/3*a^9 + 1/3*a^7 + 1/3*a^5 - 1/3*a, 1/3*a^16 + 1/3*a^8 + 1/3, 1/6*a^17 - 1/6*a^15 - 1/6*a^13 - 1/6*a^9 + 1/3*a^5 + 1/3*a^3 + 1/6*a, 1/12*a^18 - 1/12*a^16 - 1/12*a^14 - 1/6*a^12 + 1/4*a^10 - 1/2*a^8 - 1/6*a^6 - 1/3*a^4 + 5/12*a^2 + 1/3, 1/24*a^19 - 1/24*a^17 - 1/24*a^15 - 1/12*a^13 + 1/8*a^11 - 1/4*a^9 - 1/12*a^7 + 1/3*a^5 - 7/24*a^3 - 1/3*a, 1/48*a^20 - 1/48*a^18 - 1/48*a^16 - 1/24*a^14 - 5/48*a^12 + 5/24*a^10 + 11/24*a^8 - 1/6*a^6 + 17/48*a^4 + 1/6*a^2 + 1/3, 1/96*a^21 - 1/96*a^19 - 1/96*a^17 + 7/48*a^15 - 5/96*a^13 + 7/16*a^11 - 7/16*a^9 - 5/12*a^7 + 11/32*a^5 - 5/12*a^3, 1/192*a^22 - 1/192*a^20 - 1/192*a^18 + 7/96*a^16 - 5/192*a^14 + 5/96*a^12 + 11/96*a^10 + 7/24*a^8 + 65/192*a^6 - 5/24*a^4 + 1/3*a^2 + 1/3, 1/384*a^23 - 1/384*a^21 - 1/384*a^19 + 7/192*a^17 - 5/384*a^15 - 9/64*a^13 + 25/64*a^11 + 7/48*a^9 + 43/128*a^7 + 19/48*a^5 - 1/2*a, 1/2304*a^24 - 1/2304*a^22 - 17/2304*a^20 + 47/1152*a^18 - 53/2304*a^16 - 107/1152*a^14 + 115/1152*a^12 + 139/288*a^10 + 673/2304*a^8 - 61/288*a^6 + 7/144*a^4 + 2/9*a^2 - 2/9, 1/4608*a^25 - 1/4608*a^23 - 17/4608*a^21 + 47/2304*a^19 - 53/4608*a^17 - 107/2304*a^15 - 269/2304*a^13 + 43/576*a^11 - 1631/4608*a^9 + 35/576*a^7 - 137/288*a^5 + 4/9*a^3 - 5/18*a, 1/9216*a^26 - 1/9216*a^24 - 17/9216*a^22 + 47/4608*a^20 - 53/9216*a^18 + 661/4608*a^16 - 269/4608*a^14 + 43/1152*a^12 + 2977/9216*a^10 - 349/1152*a^8 - 137/576*a^6 + 2/9*a^4 + 13/36*a^2 - 1/3, 1/18432*a^27 - 1/18432*a^25 - 17/18432*a^23 + 47/9216*a^21 - 53/18432*a^19 + 661/9216*a^17 - 269/9216*a^15 + 43/2304*a^13 + 2977/18432*a^11 + 803/2304*a^9 + 439/1152*a^7 + 1/9*a^5 - 23/72*a^3 - 1/6*a, 1/6746112*a^28 + 121/2248704*a^26 - 317/6746112*a^24 + 3257/3373056*a^22 + 3803/749568*a^20 - 76057/3373056*a^18 + 189871/3373056*a^16 + 1193/70272*a^14 + 89921/6746112*a^12 + 527425/1686528*a^10 + 6311/15616*a^8 - 22805/105408*a^6 + 2575/26352*a^4 - 431/2196*a^2 + 553/1647, 1/13492224*a^29 + 121/4497408*a^27 - 317/13492224*a^25 + 3257/6746112*a^23 + 3803/1499136*a^21 - 76057/6746112*a^19 + 189871/6746112*a^17 + 1193/140544*a^15 - 2158783/13492224*a^13 + 1651777/3373056*a^11 - 9305/31232*a^9 - 93077/210816*a^7 + 2575/52704*a^5 + 1033/4392*a^3 - 1643/3294*a, 1/1969864704*a^30 - 137/1969864704*a^28 + 64135/1969864704*a^26 + 18091/984932352*a^24 - 2180405/1969864704*a^22 + 5726105/984932352*a^20 + 13104481/328310784*a^18 + 4764025/246233088*a^16 + 220845377/1969864704*a^14 - 1613989/41038848*a^12 - 477989/30779136*a^10 - 447275/1923696*a^8 - 219485/3847392*a^6 - 554663/1923696*a^4 + 21421/120231*a^2 + 34819/120231, 1/3939729408*a^31 - 137/3939729408*a^29 + 64135/3939729408*a^27 + 18091/1969864704*a^25 - 2180405/3939729408*a^23 + 5726105/1969864704*a^21 + 13104481/656621568*a^19 + 4764025/492466176*a^17 + 220845377/3939729408*a^15 + 12065627/82077696*a^13 - 20997413/61558272*a^11 + 1476421/3847392*a^9 + 2345443/7694784*a^7 + 1369033/3847392*a^5 - 58733/240462*a^3 + 37448/120231*a

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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{125015}{1969864704} a^{31} + \frac{1642879}{1969864704} a^{29} + \frac{8427127}{1969864704} a^{27} + \frac{7974151}{984932352} a^{25} + \frac{4425691}{1969864704} a^{23} - \frac{20003167}{984932352} a^{21} - \frac{14819299}{328310784} a^{19} + \frac{426463}{246233088} a^{17} + \frac{250786073}{1969864704} a^{15} + \frac{291593}{3419904} a^{13} - \frac{158447311}{246233088} a^{11} - \frac{45748283}{30779136} a^{9} - \frac{4264409}{15389568} a^{7} + \frac{27712013}{3847392} a^{5} + \frac{17617691}{961848} a^{3} + \frac{4364753}{240462} a $ -(125015)/(1969864704)a^(31) + (1642879)/(1969864704)a^(29) + (8427127)/(1969864704)a^(27) + (7974151)/(984932352)a^(25) + (4425691)/(1969864704)a^(23) - (20003167)/(984932352)a^(21) - (14819299)/(328310784)a^(19) + (426463)/(246233088)a^(17) + (250786073)/(1969864704)a^(15) + (291593)/(3419904)a^(13) - (158447311)/(246233088)a^(11) - (45748283)/(30779136)a^(9) - (4264409)/(15389568)a^(7) + (27712013)/(3847392)a^(5) + (17617691)/(961848)a^(3) + (4364753)/(240462)a (order $12$)	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:	$\frac{125015}{1969864704}a^{31}-\frac{1642879}{1969864704}a^{29}-\frac{8427127}{1969864704}a^{27}-\frac{7974151}{984932352}a^{25}-\frac{4425691}{1969864704}a^{23}+\frac{20003167}{984932352}a^{21}+\frac{14819299}{328310784}a^{19}-\frac{426463}{246233088}a^{17}-\frac{250786073}{1969864704}a^{15}-\frac{291593}{3419904}a^{13}+\frac{158447311}{246233088}a^{11}+\frac{45748283}{30779136}a^{9}+\frac{4264409}{15389568}a^{7}-\frac{27712013}{3847392}a^{5}-\frac{17617691}{961848}a^{3}-\frac{4364753}{240462}a+1$, $\frac{15307}{13492224}a^{30}+\frac{40033}{13492224}a^{28}-\frac{1919}{13492224}a^{26}-\frac{93029}{6746112}a^{24}-\frac{351815}{13492224}a^{22}-\frac{39919}{6746112}a^{20}+\frac{22731}{249856}a^{18}+\frac{50665}{421632}a^{16}-\frac{2694853}{13492224}a^{14}-\frac{926465}{1124352}a^{12}-\frac{357757}{1686528}a^{10}+\frac{625183}{210816}a^{8}+\frac{879691}{105408}a^{6}+\frac{138263}{26352}a^{4}-\frac{108703}{6588}a^{2}-\frac{54463}{1647}$, $\frac{331849}{1313243136}a^{31}-\frac{19572127}{3939729408}a^{29}-\frac{3507253}{145915904}a^{27}-\frac{87419777}{1969864704}a^{25}-\frac{40122175}{3939729408}a^{23}+\frac{75659963}{656621568}a^{21}+\frac{493193365}{1969864704}a^{19}-\frac{5292779}{246233088}a^{17}-\frac{923076695}{1313243136}a^{15}-\frac{364232111}{984932352}a^{13}+\frac{446566981}{123116544}a^{11}+\frac{82967927}{10259712}a^{9}+\frac{9159731}{15389568}a^{7}-\frac{19824899}{480924}a^{5}-\frac{10893079}{106872}a^{3}-\frac{11870576}{120231}a$, $\frac{2882689}{1969864704}a^{30}+\frac{16600055}{1969864704}a^{28}+\frac{39722695}{1969864704}a^{26}+\frac{5808377}{328310784}a^{24}-\frac{55205557}{1969864704}a^{22}-\frac{103592647}{984932352}a^{20}-\frac{79603837}{984932352}a^{18}+\frac{45479297}{246233088}a^{16}+\frac{605258177}{1969864704}a^{14}-\frac{107661895}{123116544}a^{12}-\frac{51732167}{15389568}a^{10}-\frac{82037399}{30779136}a^{8}+\frac{3233963}{284992}a^{6}+\frac{40339909}{961848}a^{4}+\frac{30153451}{480924}a^{2}+\frac{4475792}{120231}$, $\frac{1433}{26984448}a^{30}+\frac{11615}{26984448}a^{28}+\frac{42239}{26984448}a^{26}+\frac{24491}{13492224}a^{24}-\frac{22909}{26984448}a^{22}-\frac{100127}{13492224}a^{20}-\frac{41215}{4497408}a^{18}+\frac{24509}{3373056}a^{16}+\frac{979897}{26984448}a^{14}-\frac{16529}{562176}a^{12}-\frac{421723}{1686528}a^{10}-\frac{131959}{421632}a^{8}+\frac{48061}{105408}a^{6}+\frac{68837}{26352}a^{4}+\frac{18157}{3294}a^{2}+\frac{5666}{1647}$, $\frac{317671}{218873856}a^{30}+\frac{7959391}{656621568}a^{28}+\frac{23323783}{656621568}a^{26}+\frac{14208673}{328310784}a^{24}-\frac{5732395}{218873856}a^{22}-\frac{19757075}{109436928}a^{20}-\frac{72335965}{328310784}a^{18}+\frac{3087593}{13679616}a^{16}+\frac{52579213}{72957952}a^{14}-\frac{130788733}{164155392}a^{12}-\frac{78826535}{13679616}a^{10}-\frac{12486361}{1709952}a^{8}+\frac{16062065}{1282464}a^{6}+\frac{11116919}{160308}a^{4}+\frac{20022257}{160308}a^{2}+\frac{411542}{4453}$, $\frac{1304891}{1969864704}a^{30}+\frac{1430191}{656621568}a^{28}+\frac{3846077}{1969864704}a^{26}-\frac{4025063}{984932352}a^{24}-\frac{3069439}{218873856}a^{22}-\frac{12499373}{984932352}a^{20}+\frac{30440273}{984932352}a^{18}+\frac{5738113}{82077696}a^{16}-\frac{131329541}{1969864704}a^{14}-\frac{55378387}{123116544}a^{12}-\frac{258083}{569984}a^{10}+\frac{32562305}{30779136}a^{8}+\frac{9189521}{1923696}a^{6}+\frac{356893}{53436}a^{4}-\frac{277615}{480924}a^{2}-\frac{133480}{13359}$, $\frac{4103}{13492224}a^{30}+\frac{6215}{13492224}a^{28}-\frac{21421}{13492224}a^{26}-\frac{22013}{3373056}a^{24}-\frac{107431}{13492224}a^{22}+\frac{4313}{843264}a^{20}+\frac{10397}{249856}a^{18}+\frac{126187}{3373056}a^{16}-\frac{1312313}{13492224}a^{14}-\frac{568727}{2248704}a^{12}+\frac{270991}{1686528}a^{10}+\frac{543985}{421632}a^{8}+\frac{118747}{52704}a^{6}-\frac{4075}{3294}a^{4}-\frac{18116}{1647}a^{2}-\frac{27002}{1647}$, $\frac{5640785}{3939729408}a^{31}+\frac{1535795}{656621568}a^{30}+\frac{3759575}{437747712}a^{29}+\frac{21280943}{1969864704}a^{28}+\frac{83051087}{3939729408}a^{27}+\frac{13452133}{656621568}a^{26}+\frac{38590183}{1969864704}a^{25}+\frac{7054339}{984932352}a^{24}-\frac{35383687}{1313243136}a^{23}-\frac{95943709}{1969864704}a^{22}-\frac{3554039}{32292864}a^{21}-\frac{36409357}{328310784}a^{20}-\frac{179296933}{1969864704}a^{19}-\frac{8130605}{984932352}a^{18}+\frac{3378565}{18239488}a^{17}+\frac{69595049}{246233088}a^{16}+\frac{1324424785}{3939729408}a^{15}+\frac{111870739}{656621568}a^{14}-\frac{418769921}{492466176}a^{13}-\frac{93274759}{61558272}a^{12}-\frac{4505167}{1282464}a^{11}-\frac{445348123}{123116544}a^{10}-\frac{45963679}{15389568}a^{9}-\frac{4293671}{10259712}a^{8}+\frac{172937615}{15389568}a^{7}+\frac{137887793}{7694784}a^{6}+\frac{13989317}{320616}a^{5}+\frac{89567075}{1923696}a^{4}+\frac{32268829}{480924}a^{3}+\frac{2046935}{40077}a^{2}+\frac{3267649}{80154}a+\frac{1422863}{120231}$, $\frac{274373}{437747712}a^{31}+\frac{111823}{218873856}a^{30}+\frac{14199323}{3939729408}a^{29}+\frac{7837249}{1969864704}a^{28}+\frac{11107513}{1313243136}a^{27}+\frac{7586459}{656621568}a^{26}+\frac{13206247}{1969864704}a^{25}+\frac{13726397}{984932352}a^{24}-\frac{50458705}{3939729408}a^{23}-\frac{16918355}{1969864704}a^{22}-\frac{3244169}{72957952}a^{21}-\frac{6338249}{109436928}a^{20}-\frac{55597049}{1969864704}a^{19}-\frac{71042203}{984932352}a^{18}+\frac{40724537}{492466176}a^{17}+\frac{17258359}{246233088}a^{16}+\frac{157874255}{1313243136}a^{15}+\frac{148603885}{656621568}a^{14}-\frac{24475031}{61558272}a^{13}-\frac{31735957}{123116544}a^{12}-\frac{174423773}{123116544}a^{11}-\frac{112963345}{61558272}a^{10}-\frac{6727183}{6839808}a^{9}-\frac{3997429}{1709952}a^{8}+\frac{78294473}{15389568}a^{7}+\frac{31904479}{7694784}a^{6}+\frac{67808147}{3847392}a^{5}+\frac{43083463}{1923696}a^{4}+\frac{2723293}{106872}a^{3}+\frac{1091321}{26718}a^{2}+\frac{3150197}{240462}a+\frac{3707758}{120231}$, $\frac{1896289}{3939729408}a^{31}-\frac{166777}{123116544}a^{30}+\frac{4645067}{3939729408}a^{29}-\frac{312575}{41038848}a^{28}-\frac{1198877}{3939729408}a^{27}-\frac{1084133}{61558272}a^{26}-\frac{4089005}{656621568}a^{25}-\frac{1711483}{123116544}a^{24}-\frac{44268589}{3939729408}a^{23}+\frac{561349}{20519424}a^{22}-\frac{3794425}{1969864704}a^{21}+\frac{5678761}{61558272}a^{20}+\frac{81901103}{1969864704}a^{19}+\frac{7226755}{123116544}a^{18}+\frac{6616433}{123116544}a^{17}-\frac{399975}{2279936}a^{16}-\frac{340012639}{3939729408}a^{15}-\frac{30469847}{123116544}a^{14}-\frac{349957231}{984932352}a^{13}+\frac{51823753}{61558272}a^{12}-\frac{349291}{4036608}a^{11}+\frac{120730301}{41038848}a^{10}+\frac{19678903}{15389568}a^{9}+\frac{58941239}{30779136}a^{8}+\frac{4420133}{1282464}a^{7}-\frac{83169475}{7694784}a^{6}+\frac{6983249}{3847392}a^{5}-\frac{23687873}{641232}a^{4}-\frac{3828343}{480924}a^{3}-\frac{25209781}{480924}a^{2}-\frac{1791242}{120231}a-\frac{1100725}{40077}$, $\frac{106297}{328310784}a^{31}-\frac{1119625}{246233088}a^{30}+\frac{224995}{36478976}a^{29}-\frac{13156565}{492466176}a^{28}+\frac{122299}{5382144}a^{27}-\frac{31702583}{492466176}a^{26}+\frac{5760751}{164155392}a^{25}-\frac{28129813}{492466176}a^{24}-\frac{916081}{328310784}a^{23}+\frac{5431811}{61558272}a^{22}-\frac{18410297}{164155392}a^{21}+\frac{165376735}{492466176}a^{20}-\frac{10432069}{54718464}a^{19}+\frac{21072469}{82077696}a^{18}+\frac{3218099}{41038848}a^{17}-\frac{145666781}{246233088}a^{16}+\frac{185605729}{328310784}a^{15}-\frac{244737485}{246233088}a^{14}-\frac{476709}{9119744}a^{13}+\frac{449203139}{164155392}a^{12}-\frac{290125609}{82077696}a^{11}+\frac{663229973}{61558272}a^{10}-\frac{127659667}{20519424}a^{9}+\frac{263328325}{30779136}a^{8}+\frac{18475325}{5129856}a^{7}-\frac{276711301}{7694784}a^{6}+\frac{53165503}{1282464}a^{5}-\frac{32235343}{240462}a^{4}+\frac{3145863}{35624}a^{3}-\frac{48339967}{240462}a^{2}+\frac{3094799}{40077}a-\frac{14229944}{120231}$, $\frac{994057}{492466176}a^{31}+\frac{2376403}{984932352}a^{30}+\frac{13568579}{984932352}a^{29}+\frac{8573645}{984932352}a^{28}+\frac{35967317}{984932352}a^{27}+\frac{10209349}{984932352}a^{26}+\frac{38170795}{984932352}a^{25}-\frac{1780301}{164155392}a^{24}-\frac{2357179}{61558272}a^{23}-\frac{51561367}{984932352}a^{22}-\frac{184531309}{984932352}a^{21}-\frac{30956233}{492466176}a^{20}-\frac{31105517}{164155392}a^{19}+\frac{44905637}{492466176}a^{18}+\frac{138840299}{492466176}a^{17}+\frac{33476483}{123116544}a^{16}+\frac{323181161}{492466176}a^{15}-\frac{118746205}{984932352}a^{14}-\frac{42444353}{36478976}a^{13}-\frac{203924363}{123116544}a^{12}-\frac{1481777491}{246233088}a^{11}-\frac{256115033}{123116544}a^{10}-\frac{191558213}{30779136}a^{9}+\frac{5515535}{1923696}a^{8}+\frac{252336631}{15389568}a^{7}+\frac{15404861}{854976}a^{6}+\frac{142014865}{1923696}a^{5}+\frac{56829961}{1923696}a^{4}+\frac{29137301}{240462}a^{3}+\frac{4294607}{480924}a^{2}+\frac{9869107}{120231}a-\frac{3475739}{120231}$, $\frac{866651}{492466176}a^{31}-\frac{319589}{656621568}a^{30}+\frac{2836093}{328310784}a^{29}+\frac{819397}{656621568}a^{28}+\frac{17280661}{984932352}a^{27}+\frac{7056677}{656621568}a^{26}+\frac{8888959}{984932352}a^{25}+\frac{2628319}{109436928}a^{24}-\frac{23345}{641232}a^{23}+\frac{910417}{72957952}a^{22}-\frac{93314213}{984932352}a^{21}-\frac{15904585}{328310784}a^{20}-\frac{12809321}{492466176}a^{19}-\frac{4995215}{36478976}a^{18}+\frac{36042493}{164155392}a^{17}-\frac{247299}{9119744}a^{16}+\frac{86191999}{492466176}a^{15}+\frac{236942939}{656621568}a^{14}-\frac{1134146147}{984932352}a^{13}+\frac{11132399}{27359232}a^{12}-\frac{62530759}{20519424}a^{11}-\frac{10347481}{6839808}a^{10}-\frac{57076505}{61558272}a^{9}-\frac{186995}{42048}a^{8}+\frac{105509291}{7694784}a^{7}-\frac{960173}{320616}a^{6}+\frac{49710419}{1282464}a^{5}+\frac{10501151}{641232}a^{4}+\frac{5622404}{120231}a^{3}+\frac{2677099}{53436}a^{2}+\frac{148877}{8906}a+\frac{727247}{13359}$, $\frac{2219269}{3939729408}a^{31}-\frac{2998487}{1969864704}a^{30}+\frac{18408739}{3939729408}a^{29}-\frac{16726525}{1969864704}a^{28}+\frac{52259587}{3939729408}a^{27}-\frac{622649}{32292864}a^{26}+\frac{30875767}{1969864704}a^{25}-\frac{571349}{36478976}a^{24}-\frac{42754841}{3939729408}a^{23}+\frac{60095771}{1969864704}a^{22}-\frac{133784923}{1969864704}a^{21}+\frac{100052447}{984932352}a^{20}-\frac{5729403}{72957952}a^{19}+\frac{64771967}{984932352}a^{18}+\frac{42939661}{492466176}a^{17}-\frac{3040429}{15389568}a^{16}+\frac{1055669093}{3939729408}a^{15}-\frac{541733047}{1969864704}a^{14}-\frac{13776277}{41038848}a^{13}+\frac{463001633}{492466176}a^{12}-\frac{531305993}{246233088}a^{11}+\frac{201631699}{61558272}a^{10}-\frac{160940873}{61558272}a^{9}+\frac{547433}{240462}a^{8}+\frac{38159989}{7694784}a^{7}-\frac{2561563}{213744}a^{6}+\frac{100809313}{3847392}a^{5}-\frac{79751881}{1923696}a^{4}+\frac{43801897}{961848}a^{3}-\frac{28148657}{480924}a^{2}+\frac{3967043}{120231}a-\frac{3858610}{120231}$ 125015/1969864704a^31 - 1642879/1969864704a^29 - 8427127/1969864704a^27 - 7974151/984932352a^25 - 4425691/1969864704a^23 + 20003167/984932352a^21 + 14819299/328310784a^19 - 426463/246233088a^17 - 250786073/1969864704a^15 - 291593/3419904a^13 + 158447311/246233088a^11 + 45748283/30779136a^9 + 4264409/15389568a^7 - 27712013/3847392a^5 - 17617691/961848a^3 - 4364753/240462a + 1, 15307/13492224a^30 + 40033/13492224a^28 - 1919/13492224a^26 - 93029/6746112a^24 - 351815/13492224a^22 - 39919/6746112a^20 + 22731/249856a^18 + 50665/421632a^16 - 2694853/13492224a^14 - 926465/1124352a^12 - 357757/1686528a^10 + 625183/210816a^8 + 879691/105408a^6 + 138263/26352a^4 - 108703/6588a^2 - 54463/1647, 331849/1313243136a^31 - 19572127/3939729408a^29 - 3507253/145915904a^27 - 87419777/1969864704a^25 - 40122175/3939729408a^23 + 75659963/656621568a^21 + 493193365/1969864704a^19 - 5292779/246233088a^17 - 923076695/1313243136a^15 - 364232111/984932352a^13 + 446566981/123116544a^11 + 82967927/10259712a^9 + 9159731/15389568a^7 - 19824899/480924a^5 - 10893079/106872a^3 - 11870576/120231a, 2882689/1969864704a^30 + 16600055/1969864704a^28 + 39722695/1969864704a^26 + 5808377/328310784a^24 - 55205557/1969864704a^22 - 103592647/984932352a^20 - 79603837/984932352a^18 + 45479297/246233088a^16 + 605258177/1969864704a^14 - 107661895/123116544a^12 - 51732167/15389568a^10 - 82037399/30779136a^8 + 3233963/284992a^6 + 40339909/961848a^4 + 30153451/480924a^2 + 4475792/120231, 1433/26984448a^30 + 11615/26984448a^28 + 42239/26984448a^26 + 24491/13492224a^24 - 22909/26984448a^22 - 100127/13492224a^20 - 41215/4497408a^18 + 24509/3373056a^16 + 979897/26984448a^14 - 16529/562176a^12 - 421723/1686528a^10 - 131959/421632a^8 + 48061/105408a^6 + 68837/26352a^4 + 18157/3294a^2 + 5666/1647, 317671/218873856a^30 + 7959391/656621568a^28 + 23323783/656621568a^26 + 14208673/328310784a^24 - 5732395/218873856a^22 - 19757075/109436928a^20 - 72335965/328310784a^18 + 3087593/13679616a^16 + 52579213/72957952a^14 - 130788733/164155392a^12 - 78826535/13679616a^10 - 12486361/1709952a^8 + 16062065/1282464a^6 + 11116919/160308a^4 + 20022257/160308a^2 + 411542/4453, 1304891/1969864704a^30 + 1430191/656621568a^28 + 3846077/1969864704a^26 - 4025063/984932352a^24 - 3069439/218873856a^22 - 12499373/984932352a^20 + 30440273/984932352a^18 + 5738113/82077696a^16 - 131329541/1969864704a^14 - 55378387/123116544a^12 - 258083/569984a^10 + 32562305/30779136a^8 + 9189521/1923696a^6 + 356893/53436a^4 - 277615/480924a^2 - 133480/13359, 4103/13492224a^30 + 6215/13492224a^28 - 21421/13492224a^26 - 22013/3373056a^24 - 107431/13492224a^22 + 4313/843264a^20 + 10397/249856a^18 + 126187/3373056a^16 - 1312313/13492224a^14 - 568727/2248704a^12 + 270991/1686528a^10 + 543985/421632a^8 + 118747/52704a^6 - 4075/3294a^4 - 18116/1647a^2 - 27002/1647, 5640785/3939729408a^31 + 1535795/656621568a^30 + 3759575/437747712a^29 + 21280943/1969864704a^28 + 83051087/3939729408a^27 + 13452133/656621568a^26 + 38590183/1969864704a^25 + 7054339/984932352a^24 - 35383687/1313243136a^23 - 95943709/1969864704a^22 - 3554039/32292864a^21 - 36409357/328310784a^20 - 179296933/1969864704a^19 - 8130605/984932352a^18 + 3378565/18239488a^17 + 69595049/246233088a^16 + 1324424785/3939729408a^15 + 111870739/656621568a^14 - 418769921/492466176a^13 - 93274759/61558272a^12 - 4505167/1282464a^11 - 445348123/123116544a^10 - 45963679/15389568a^9 - 4293671/10259712a^8 + 172937615/15389568a^7 + 137887793/7694784a^6 + 13989317/320616a^5 + 89567075/1923696a^4 + 32268829/480924a^3 + 2046935/40077a^2 + 3267649/80154a + 1422863/120231, 274373/437747712a^31 + 111823/218873856a^30 + 14199323/3939729408a^29 + 7837249/1969864704a^28 + 11107513/1313243136a^27 + 7586459/656621568a^26 + 13206247/1969864704a^25 + 13726397/984932352a^24 - 50458705/3939729408a^23 - 16918355/1969864704a^22 - 3244169/72957952a^21 - 6338249/109436928a^20 - 55597049/1969864704a^19 - 71042203/984932352a^18 + 40724537/492466176a^17 + 17258359/246233088a^16 + 157874255/1313243136a^15 + 148603885/656621568a^14 - 24475031/61558272a^13 - 31735957/123116544a^12 - 174423773/123116544a^11 - 112963345/61558272a^10 - 6727183/6839808a^9 - 3997429/1709952a^8 + 78294473/15389568a^7 + 31904479/7694784a^6 + 67808147/3847392a^5 + 43083463/1923696a^4 + 2723293/106872a^3 + 1091321/26718a^2 + 3150197/240462a + 3707758/120231, 1896289/3939729408a^31 - 166777/123116544a^30 + 4645067/3939729408a^29 - 312575/41038848a^28 - 1198877/3939729408a^27 - 1084133/61558272a^26 - 4089005/656621568a^25 - 1711483/123116544a^24 - 44268589/3939729408a^23 + 561349/20519424a^22 - 3794425/1969864704a^21 + 5678761/61558272a^20 + 81901103/1969864704a^19 + 7226755/123116544a^18 + 6616433/123116544a^17 - 399975/2279936a^16 - 340012639/3939729408a^15 - 30469847/123116544a^14 - 349957231/984932352a^13 + 51823753/61558272a^12 - 349291/4036608a^11 + 120730301/41038848a^10 + 19678903/15389568a^9 + 58941239/30779136a^8 + 4420133/1282464a^7 - 83169475/7694784a^6 + 6983249/3847392a^5 - 23687873/641232a^4 - 3828343/480924a^3 - 25209781/480924a^2 - 1791242/120231a - 1100725/40077, 106297/328310784a^31 - 1119625/246233088a^30 + 224995/36478976a^29 - 13156565/492466176a^28 + 122299/5382144a^27 - 31702583/492466176a^26 + 5760751/164155392a^25 - 28129813/492466176a^24 - 916081/328310784a^23 + 5431811/61558272a^22 - 18410297/164155392a^21 + 165376735/492466176a^20 - 10432069/54718464a^19 + 21072469/82077696a^18 + 3218099/41038848a^17 - 145666781/246233088a^16 + 185605729/328310784a^15 - 244737485/246233088a^14 - 476709/9119744a^13 + 449203139/164155392a^12 - 290125609/82077696a^11 + 663229973/61558272a^10 - 127659667/20519424a^9 + 263328325/30779136a^8 + 18475325/5129856a^7 - 276711301/7694784a^6 + 53165503/1282464a^5 - 32235343/240462a^4 + 3145863/35624a^3 - 48339967/240462a^2 + 3094799/40077a - 14229944/120231, 994057/492466176a^31 + 2376403/984932352a^30 + 13568579/984932352a^29 + 8573645/984932352a^28 + 35967317/984932352a^27 + 10209349/984932352a^26 + 38170795/984932352a^25 - 1780301/164155392a^24 - 2357179/61558272a^23 - 51561367/984932352a^22 - 184531309/984932352a^21 - 30956233/492466176a^20 - 31105517/164155392a^19 + 44905637/492466176a^18 + 138840299/492466176a^17 + 33476483/123116544a^16 + 323181161/492466176a^15 - 118746205/984932352a^14 - 42444353/36478976a^13 - 203924363/123116544a^12 - 1481777491/246233088a^11 - 256115033/123116544a^10 - 191558213/30779136a^9 + 5515535/1923696a^8 + 252336631/15389568a^7 + 15404861/854976a^6 + 142014865/1923696a^5 + 56829961/1923696a^4 + 29137301/240462a^3 + 4294607/480924a^2 + 9869107/120231a - 3475739/120231, 866651/492466176a^31 - 319589/656621568a^30 + 2836093/328310784a^29 + 819397/656621568a^28 + 17280661/984932352a^27 + 7056677/656621568a^26 + 8888959/984932352a^25 + 2628319/109436928a^24 - 23345/641232a^23 + 910417/72957952a^22 - 93314213/984932352a^21 - 15904585/328310784a^20 - 12809321/492466176a^19 - 4995215/36478976a^18 + 36042493/164155392a^17 - 247299/9119744a^16 + 86191999/492466176a^15 + 236942939/656621568a^14 - 1134146147/984932352a^13 + 11132399/27359232a^12 - 62530759/20519424a^11 - 10347481/6839808a^10 - 57076505/61558272a^9 - 186995/42048a^8 + 105509291/7694784a^7 - 960173/320616a^6 + 49710419/1282464a^5 + 10501151/641232a^4 + 5622404/120231a^3 + 2677099/53436a^2 + 148877/8906a + 727247/13359, 2219269/3939729408a^31 - 2998487/1969864704a^30 + 18408739/3939729408a^29 - 16726525/1969864704a^28 + 52259587/3939729408a^27 - 622649/32292864a^26 + 30875767/1969864704a^25 - 571349/36478976a^24 - 42754841/3939729408a^23 + 60095771/1969864704a^22 - 133784923/1969864704a^21 + 100052447/984932352a^20 - 5729403/72957952a^19 + 64771967/984932352a^18 + 42939661/492466176a^17 - 3040429/15389568a^16 + 1055669093/3939729408a^15 - 541733047/1969864704a^14 - 13776277/41038848a^13 + 463001633/492466176a^12 - 531305993/246233088a^11 + 201631699/61558272a^10 - 160940873/61558272a^9 + 547433/240462a^8 + 38159989/7694784a^7 - 2561563/213744a^6 + 100809313/3847392a^5 - 79751881/1923696a^4 + 43801897/961848a^3 - 28148657/480924a^2 + 3967043/120231*a - 3858610/120231 (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:	$ 382220804260.19714 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 382220804260.19714 \cdot 18}{12\cdot\sqrt{111539154030155436264757885742310633531656776974336}}\cr\approx \mathstrut & 0.320309486822401 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536)

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 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536, 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 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536);

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 + 7*x^30 + 23*x^28 + 38*x^26 + 11*x^24 - 94*x^22 - 186*x^20 - 24*x^18 + 385*x^16 - 96*x^14 - 2976*x^12 - 6016*x^10 + 2816*x^8 + 38912*x^6 + 94208*x^4 + 114688*x^2 + 65536);

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

$C_2^6:S_4$ (as 32T96908):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 1536

The 80 conjugacy class representatives for $C_2^6:S_4$

Character table for $C_2^6:S_4$

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{3}) $, 4.4.3981.1, $\Q(\zeta_{12})$, 8.0.361089057024.1, 8.8.36514623744.1, 8.0.12694537161.1, 8.0.4057180416.1, 8.8.1410504129.1, 8.0.142635249.1, 8.8.3249801513216.1, 16.0.10561209875301003422662656.2, 16.0.130385307102481523736576.1, 16.0.10561209875301003422662656.3, 16.0.1333317747165888577536.1, 16.16.10561209875301003422662656.1, 16.0.10561209875301003422662656.1, 16.0.161151273732009939921.1

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]

Sibling fields

Degree 32 siblings:	data not computed
Minimal sibling:	not computed

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.6.0.1}{6} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$	${\href{/padicField/7.8.0.1}{8} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	${\href{/padicField/13.3.0.1}{3} }^{8}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.8.0.1}{8} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$	${\href{/padicField/29.8.0.1}{8} }^{4}$	${\href{/padicField/31.8.0.1}{8} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$	${\href{/padicField/41.6.0.1}{6} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$	${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$3$ 3	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$89$ 89	89.2.0.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	89.2.0.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	89.2.0.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	89.2.0.1	$x^{2} + 82 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$1327$ 1327		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more