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Properties

Label 32.0.323...000.2
Degree $32$
Signature $[0, 16]$
Discriminant $3.234\times 10^{56}$
Root discriminant \(58.34\)
Ramified primes $2,3,5,29,769$
Class number not computed
Class group not computed
Galois group $D_4^2:C_2^3$ (as 32T12882)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256)
 
gp: K = bnfinit(y^32 - 12*y^31 + 102*y^30 - 632*y^29 + 3259*y^28 - 14196*y^27 + 54054*y^26 - 181168*y^25 + 538850*y^24 - 1421372*y^23 + 3301686*y^22 - 6656356*y^21 + 11242330*y^20 - 14675868*y^19 + 10736660*y^18 + 10164980*y^17 - 54031327*y^16 + 110396848*y^15 - 138789976*y^14 + 83526288*y^13 + 68196328*y^12 - 240215264*y^11 + 278975616*y^10 - 118651264*y^9 - 65191888*y^8 + 93478528*y^7 - 24294144*y^6 - 16443648*y^5 + 13102720*y^4 - 3280384*y^3 + 364032*y^2 - 15360*y + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256)
 

\( x^{32} - 12 x^{31} + 102 x^{30} - 632 x^{29} + 3259 x^{28} - 14196 x^{27} + 54054 x^{26} - 181168 x^{25} + \cdots + 256 \) Copy content Toggle raw display

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:   \(323436686508763086889139574016939351080960000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 769^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(58.34\)
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:  $2^{3/2}3^{1/2}5^{1/2}29^{1/2}769^{1/2}\approx 1635.8850815384312$
Ramified primes:   \(2\), \(3\), \(5\), \(29\), \(769\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{8}a^{20}-\frac{1}{8}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{22}+\frac{3}{16}a^{18}-\frac{1}{4}a^{16}+\frac{1}{8}a^{14}-\frac{1}{4}a^{13}+\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{23}+\frac{3}{16}a^{19}-\frac{1}{4}a^{17}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}+\frac{1}{8}a^{13}-\frac{1}{4}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{64}a^{24}-\frac{1}{32}a^{22}-\frac{1}{64}a^{20}-\frac{5}{32}a^{18}+\frac{1}{8}a^{17}+\frac{7}{32}a^{16}+\frac{3}{16}a^{15}-\frac{1}{32}a^{14}+\frac{1}{16}a^{13}+\frac{1}{32}a^{12}+\frac{3}{16}a^{11}+\frac{3}{16}a^{10}+\frac{5}{16}a^{9}+\frac{25}{64}a^{8}-\frac{5}{16}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{64}a^{25}-\frac{1}{32}a^{23}-\frac{1}{64}a^{21}-\frac{5}{32}a^{19}+\frac{1}{8}a^{18}+\frac{7}{32}a^{17}+\frac{3}{16}a^{16}-\frac{1}{32}a^{15}+\frac{1}{16}a^{14}+\frac{1}{32}a^{13}+\frac{3}{16}a^{12}+\frac{3}{16}a^{11}-\frac{3}{16}a^{10}+\frac{25}{64}a^{9}+\frac{3}{16}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{16}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{128}a^{26}+\frac{3}{128}a^{22}+\frac{1}{32}a^{20}-\frac{3}{16}a^{19}+\frac{9}{64}a^{18}-\frac{1}{32}a^{17}+\frac{5}{64}a^{16}+\frac{7}{32}a^{15}+\frac{7}{64}a^{14}+\frac{5}{32}a^{13}-\frac{1}{4}a^{12}+\frac{3}{32}a^{11}+\frac{1}{128}a^{10}+\frac{13}{32}a^{9}+\frac{9}{64}a^{8}+\frac{1}{16}a^{7}-\frac{7}{32}a^{6}+\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{128}a^{27}+\frac{3}{128}a^{23}+\frac{1}{32}a^{21}-\frac{1}{16}a^{20}+\frac{9}{64}a^{19}-\frac{1}{32}a^{18}+\frac{5}{64}a^{17}+\frac{3}{32}a^{16}+\frac{7}{64}a^{15}+\frac{5}{32}a^{14}-\frac{1}{4}a^{13}-\frac{5}{32}a^{12}+\frac{1}{128}a^{11}+\frac{5}{32}a^{10}+\frac{9}{64}a^{9}+\frac{5}{16}a^{8}+\frac{9}{32}a^{7}-\frac{7}{16}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{512}a^{28}-\frac{1}{512}a^{24}+\frac{3}{128}a^{22}-\frac{3}{64}a^{21}-\frac{5}{256}a^{20}-\frac{1}{128}a^{19}-\frac{7}{256}a^{18}-\frac{1}{128}a^{17}-\frac{5}{256}a^{16}-\frac{7}{128}a^{15}-\frac{11}{64}a^{14}+\frac{15}{128}a^{13}-\frac{71}{512}a^{12}-\frac{15}{128}a^{11}+\frac{49}{256}a^{10}+\frac{23}{64}a^{9}-\frac{1}{2}a^{8}-\frac{11}{32}a^{7}+\frac{1}{64}a^{6}+\frac{3}{16}a^{5}-\frac{3}{8}a^{4}-\frac{3}{8}a^{3}+\frac{5}{16}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{6656}a^{29}-\frac{5}{6656}a^{28}+\frac{1}{1664}a^{27}+\frac{5}{1664}a^{26}-\frac{49}{6656}a^{25}-\frac{3}{6656}a^{24}+\frac{15}{832}a^{23}-\frac{9}{832}a^{22}+\frac{87}{3328}a^{21}+\frac{147}{3328}a^{20}-\frac{601}{3328}a^{19}+\frac{341}{3328}a^{18}-\frac{383}{3328}a^{17}+\frac{591}{3328}a^{16}-\frac{409}{1664}a^{15}-\frac{373}{1664}a^{14}+\frac{29}{6656}a^{13}-\frac{249}{6656}a^{12}-\frac{15}{3328}a^{11}-\frac{631}{3328}a^{10}-\frac{153}{416}a^{9}-\frac{163}{416}a^{8}-\frac{151}{832}a^{7}+\frac{281}{832}a^{6}-\frac{37}{104}a^{5}-\frac{33}{104}a^{4}-\frac{23}{208}a^{3}+\frac{33}{208}a^{2}-\frac{17}{104}a-\frac{3}{104}$, $\frac{1}{59\!\cdots\!24}a^{30}+\frac{99\!\cdots\!27}{14\!\cdots\!56}a^{29}+\frac{35\!\cdots\!03}{14\!\cdots\!56}a^{28}+\frac{49\!\cdots\!65}{14\!\cdots\!64}a^{27}+\frac{14\!\cdots\!07}{59\!\cdots\!24}a^{26}+\frac{10\!\cdots\!93}{14\!\cdots\!56}a^{25}-\frac{16\!\cdots\!41}{36\!\cdots\!64}a^{24}-\frac{16\!\cdots\!09}{73\!\cdots\!28}a^{23}-\frac{47\!\cdots\!49}{29\!\cdots\!12}a^{22}+\frac{75\!\cdots\!67}{13\!\cdots\!96}a^{21}-\frac{14\!\cdots\!59}{29\!\cdots\!12}a^{20}-\frac{33\!\cdots\!03}{14\!\cdots\!56}a^{19}+\frac{47\!\cdots\!91}{29\!\cdots\!12}a^{18}-\frac{21\!\cdots\!97}{14\!\cdots\!56}a^{17}-\frac{81\!\cdots\!21}{36\!\cdots\!64}a^{16}-\frac{25\!\cdots\!41}{14\!\cdots\!56}a^{15}+\frac{10\!\cdots\!21}{59\!\cdots\!24}a^{14}-\frac{20\!\cdots\!63}{92\!\cdots\!16}a^{13}+\frac{82\!\cdots\!91}{29\!\cdots\!12}a^{12}+\frac{13\!\cdots\!69}{92\!\cdots\!16}a^{11}-\frac{54\!\cdots\!91}{36\!\cdots\!64}a^{10}-\frac{13\!\cdots\!81}{18\!\cdots\!32}a^{9}+\frac{33\!\cdots\!25}{73\!\cdots\!28}a^{8}+\frac{96\!\cdots\!11}{46\!\cdots\!08}a^{7}+\frac{74\!\cdots\!77}{46\!\cdots\!08}a^{6}+\frac{10\!\cdots\!93}{57\!\cdots\!51}a^{5}-\frac{18\!\cdots\!75}{18\!\cdots\!32}a^{4}-\frac{48\!\cdots\!31}{11\!\cdots\!02}a^{3}+\frac{37\!\cdots\!71}{92\!\cdots\!16}a^{2}-\frac{37\!\cdots\!77}{57\!\cdots\!51}a-\frac{29\!\cdots\!89}{23\!\cdots\!04}$, $\frac{1}{56\!\cdots\!48}a^{31}+\frac{15\!\cdots\!29}{21\!\cdots\!48}a^{30}+\frac{17\!\cdots\!27}{28\!\cdots\!24}a^{29}+\frac{17\!\cdots\!23}{28\!\cdots\!24}a^{28}-\frac{20\!\cdots\!57}{56\!\cdots\!48}a^{27}+\frac{44\!\cdots\!71}{28\!\cdots\!24}a^{26}-\frac{19\!\cdots\!89}{28\!\cdots\!24}a^{25}-\frac{77\!\cdots\!67}{28\!\cdots\!24}a^{24}+\frac{13\!\cdots\!75}{28\!\cdots\!24}a^{23}+\frac{62\!\cdots\!43}{35\!\cdots\!28}a^{22}+\frac{33\!\cdots\!75}{28\!\cdots\!24}a^{21}-\frac{76\!\cdots\!99}{12\!\cdots\!92}a^{20}+\frac{44\!\cdots\!25}{28\!\cdots\!24}a^{19}+\frac{10\!\cdots\!65}{14\!\cdots\!12}a^{18}+\frac{64\!\cdots\!39}{14\!\cdots\!12}a^{17}-\frac{16\!\cdots\!31}{70\!\cdots\!56}a^{16}-\frac{10\!\cdots\!43}{56\!\cdots\!48}a^{15}-\frac{40\!\cdots\!13}{28\!\cdots\!24}a^{14}+\frac{14\!\cdots\!93}{70\!\cdots\!56}a^{13}-\frac{58\!\cdots\!75}{28\!\cdots\!24}a^{12}-\frac{17\!\cdots\!73}{14\!\cdots\!12}a^{11}+\frac{41\!\cdots\!87}{14\!\cdots\!12}a^{10}+\frac{19\!\cdots\!67}{70\!\cdots\!56}a^{9}-\frac{11\!\cdots\!09}{32\!\cdots\!48}a^{8}-\frac{69\!\cdots\!89}{35\!\cdots\!28}a^{7}+\frac{63\!\cdots\!79}{35\!\cdots\!28}a^{6}+\frac{77\!\cdots\!59}{17\!\cdots\!64}a^{5}+\frac{21\!\cdots\!59}{88\!\cdots\!32}a^{4}-\frac{22\!\cdots\!25}{44\!\cdots\!16}a^{3}-\frac{25\!\cdots\!61}{80\!\cdots\!12}a^{2}+\frac{13\!\cdots\!91}{44\!\cdots\!16}a-\frac{11\!\cdots\!31}{44\!\cdots\!16}$ Copy content Toggle raw display

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

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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{9681025765132954395609012379}{21001820160311694159622526253764} a^{31} - \frac{923535258826415065700059008855}{168014561282493553276980210030112} a^{30} + \frac{3915293652427928627024401004501}{84007280641246776638490105015056} a^{29} - \frac{48362885045964593946503500783859}{168014561282493553276980210030112} a^{28} + \frac{4784784349631026151006884051171}{3231049255432568332249619423656} a^{27} - \frac{1081012437539006998332384248344747}{168014561282493553276980210030112} a^{26} + \frac{2053254201986854239326991397709735}{84007280641246776638490105015056} a^{25} - \frac{13728560205886187710358911664536403}{168014561282493553276980210030112} a^{24} + \frac{10181124954299707977784546707232353}{42003640320623388319245052507528} a^{23} - \frac{6693794386793038799421342765794619}{10500910080155847079811263126882} a^{22} + \frac{30986923415312589317394683452922723}{21001820160311694159622526253764} a^{21} - \frac{248756379802764898850548643018131067}{84007280641246776638490105015056} a^{20} + \frac{208686669009162675301410097882757531}{42003640320623388319245052507528} a^{19} - \frac{10353565880874548774627978571390879}{1615524627716284166124809711828} a^{18} + \frac{188857026423422380215683271661040195}{42003640320623388319245052507528} a^{17} + \frac{418956329796294778462154790080169683}{84007280641246776638490105015056} a^{16} - \frac{128676841135038693316754547333423783}{5250455040077923539905631563441} a^{15} + \frac{8246279233543699220649744348509358873}{168014561282493553276980210030112} a^{14} - \frac{390552445161640332406829553951507223}{6462098510865136664499238847312} a^{13} + \frac{5754415563916689593529730990234217963}{168014561282493553276980210030112} a^{12} + \frac{1414204990586334665651793434080110709}{42003640320623388319245052507528} a^{11} - \frac{1134473824859730229569827043911273705}{10500910080155847079811263126882} a^{10} + \frac{2535411012342194229646550882187632755}{21001820160311694159622526253764} a^{9} - \frac{3887966940879497299518677601440897471}{84007280641246776638490105015056} a^{8} - \frac{345083888683122150599333536676264389}{10500910080155847079811263126882} a^{7} + \frac{212400506418689484614122979845091232}{5250455040077923539905631563441} a^{6} - \frac{44176102757659574845157629152279523}{5250455040077923539905631563441} a^{5} - \frac{83983730531589073308431750284134343}{10500910080155847079811263126882} a^{4} + \frac{28490313510058955693645393399479742}{5250455040077923539905631563441} a^{3} - \frac{6073732305556373265659569088263666}{5250455040077923539905631563441} a^{2} + \frac{42615330623753120773111911208760}{403881156929071041531202427957} a - \frac{10926972916013630821050666881876}{5250455040077923539905631563441} \)  (order $6$) Copy content Toggle raw display
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot R \cdot h}{6\cdot\sqrt{323436686508763086889139574016939351080960000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256)
 
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 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256, 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 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256);
 
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 - 12*x^31 + 102*x^30 - 632*x^29 + 3259*x^28 - 14196*x^27 + 54054*x^26 - 181168*x^25 + 538850*x^24 - 1421372*x^23 + 3301686*x^22 - 6656356*x^21 + 11242330*x^20 - 14675868*x^19 + 10736660*x^18 + 10164980*x^17 - 54031327*x^16 + 110396848*x^15 - 138789976*x^14 + 83526288*x^13 + 68196328*x^12 - 240215264*x^11 + 278975616*x^10 - 118651264*x^9 - 65191888*x^8 + 93478528*x^7 - 24294144*x^6 - 16443648*x^5 + 13102720*x^4 - 3280384*x^3 + 364032*x^2 - 15360*x + 256);
 
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

$D_4^2:C_2^3$ (as 32T12882):

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 512
The 80 conjugacy class representatives for $D_4^2:C_2^3$
Character table for $D_4^2:C_2^3$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{10}) \), 4.4.725.1, 4.0.417600.1, 4.0.6525.1, 4.4.46400.1, \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{10})\), 8.8.32740655625.1, 8.0.1655626240000.1, 8.0.404205625.1, 8.8.134105725440000.1, 8.0.207360000.1, 8.0.174389760000.27, 8.0.174389760000.12, 8.0.42575625.1, 8.0.174389760000.40, 8.8.2152960000.1, 8.0.174389760000.25, 16.0.30411788392857600000000.2, 16.0.17984345595788663193600000000.1, 16.0.17984345595788663193600000000.2, 16.0.1071950530754844140625.1, 16.0.17984345595788663193600000000.5, 16.16.17984345595788663193600000000.2, 16.0.2741098246576537600000000.1

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(3\) Copy content Toggle raw display 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}$
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}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(29\) Copy content Toggle raw display 29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(769\) Copy content Toggle raw display 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$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$