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Properties

Label 32.0.87309122741...0000.6
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $92.66$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34093921, 0, 921344, 0, 4540976, 0, 605472, 0, 8338152, 0, -4206560, 0, 2598488, 0, -536128, 0, 12827, 0, -10048, 0, 12800, 0, 1760, 0, -1460, 0, -256, 0, 56, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 56*x^28 - 256*x^26 - 1460*x^24 + 1760*x^22 + 12800*x^20 - 10048*x^18 + 12827*x^16 - 536128*x^14 + 2598488*x^12 - 4206560*x^10 + 8338152*x^8 + 605472*x^6 + 4540976*x^4 + 921344*x^2 + 34093921)
 
gp: K = bnfinit(x^32 + 16*x^30 + 56*x^28 - 256*x^26 - 1460*x^24 + 1760*x^22 + 12800*x^20 - 10048*x^18 + 12827*x^16 - 536128*x^14 + 2598488*x^12 - 4206560*x^10 + 8338152*x^8 + 605472*x^6 + 4540976*x^4 + 921344*x^2 + 34093921, 1)
 

Normalized defining polynomial

\( x^{32} + 16 x^{30} + 56 x^{28} - 256 x^{26} - 1460 x^{24} + 1760 x^{22} + 12800 x^{20} - 10048 x^{18} + 12827 x^{16} - 536128 x^{14} + 2598488 x^{12} - 4206560 x^{10} + 8338152 x^{8} + 605472 x^{6} + 4540976 x^{4} + 921344 x^{2} + 34093921 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(873091227416114037923609044826021953536000000000000000000000000=2^{128}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.66$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(43,·)$, $\chi_{480}(389,·)$, $\chi_{480}(263,·)$, $\chi_{480}(407,·)$, $\chi_{480}(269,·)$, $\chi_{480}(143,·)$, $\chi_{480}(403,·)$, $\chi_{480}(149,·)$, $\chi_{480}(23,·)$, $\chi_{480}(409,·)$, $\chi_{480}(283,·)$, $\chi_{480}(29,·)$, $\chi_{480}(287,·)$, $\chi_{480}(289,·)$, $\chi_{480}(163,·)$, $\chi_{480}(167,·)$, $\chi_{480}(169,·)$, $\chi_{480}(427,·)$, $\chi_{480}(47,·)$, $\chi_{480}(49,·)$, $\chi_{480}(307,·)$, $\chi_{480}(187,·)$, $\chi_{480}(67,·)$, $\chi_{480}(461,·)$, $\chi_{480}(341,·)$, $\chi_{480}(221,·)$, $\chi_{480}(101,·)$, $\chi_{480}(361,·)$, $\chi_{480}(241,·)$, $\chi_{480}(121,·)$, $\chi_{480}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{5839} a^{23} + \frac{13}{5839} a^{21} - \frac{2500}{5839} a^{19} + \frac{2494}{5839} a^{17} - \frac{1145}{5839} a^{15} - \frac{113}{5839} a^{13} + \frac{2338}{5839} a^{11} + \frac{395}{5839} a^{9} - \frac{1119}{5839} a^{7} - \frac{189}{5839} a^{5} - \frac{2188}{5839} a^{3} - \frac{30}{5839} a$, $\frac{1}{5074091} a^{24} - \frac{1845111}{5074091} a^{22} - \frac{144079}{461281} a^{20} + \frac{878344}{5074091} a^{18} + \frac{2328616}{5074091} a^{16} + \frac{1249433}{5074091} a^{14} - \frac{1848625}{5074091} a^{12} - \frac{1354253}{5074091} a^{10} - \frac{2050608}{5074091} a^{8} - \frac{1640948}{5074091} a^{6} + \frac{307279}{5074091} a^{4} + \frac{834947}{5074091} a^{2} - \frac{58}{869}$, $\frac{1}{5074091} a^{25} - \frac{224}{5074091} a^{23} + \frac{191118}{461281} a^{21} + \frac{1009563}{5074091} a^{19} + \frac{1276257}{5074091} a^{17} - \frac{324326}{5074091} a^{15} - \frac{2283125}{5074091} a^{13} - \frac{985797}{5074091} a^{11} + \frac{1084744}{5074091} a^{9} - \frac{914464}{5074091} a^{7} + \frac{1735915}{5074091} a^{5} - \frac{1875464}{5074091} a^{3} + \frac{129729}{5074091} a$, $\frac{1}{5074091} a^{26} - \frac{201195}{5074091} a^{22} + \frac{1185277}{5074091} a^{20} + \frac{135764}{5074091} a^{18} - \frac{1345715}{5074091} a^{16} - \frac{1485138}{5074091} a^{14} + \frac{997665}{5074091} a^{12} + \frac{2177532}{5074091} a^{10} + \frac{1491625}{5074091} a^{8} - \frac{501885}{5074091} a^{6} + \frac{991849}{5074091} a^{4} - \frac{583510}{5074091} a^{2} + \frac{43}{869}$, $\frac{1}{5074091} a^{27} + \frac{413}{5074091} a^{23} - \frac{1267910}{5074091} a^{21} - \frac{1549227}{5074091} a^{19} - \frac{870372}{5074091} a^{17} + \frac{1081888}{5074091} a^{15} - \frac{1487675}{5074091} a^{13} + \frac{1646573}{5074091} a^{11} - \frac{58671}{5074091} a^{9} + \frac{2232858}{5074091} a^{7} - \frac{1593426}{5074091} a^{5} - \frac{255897}{5074091} a^{3} - \frac{723072}{5074091} a$, $\frac{1}{5074091} a^{28} - \frac{350717}{5074091} a^{22} - \frac{1556069}{5074091} a^{20} + \frac{1708108}{5074091} a^{18} - \frac{1633321}{5074091} a^{16} + \frac{53778}{5074091} a^{14} - \frac{1059043}{5074091} a^{12} + \frac{1097808}{5074091} a^{10} + \frac{1760765}{5074091} a^{8} + \frac{1263995}{5074091} a^{6} - \frac{309849}{5074091} a^{4} - \frac{517995}{5074091} a^{2} - \frac{378}{869}$, $\frac{1}{5074091} a^{29} + \frac{359}{5074091} a^{23} - \frac{2066172}{5074091} a^{21} + \frac{1835851}{5074091} a^{19} + \frac{1206571}{5074091} a^{17} - \frac{1075053}{5074091} a^{15} - \frac{137903}{5074091} a^{13} - \frac{89246}{5074091} a^{11} - \frac{1638763}{5074091} a^{9} - \frac{885042}{5074091} a^{7} - \frac{700030}{5074091} a^{5} - \frac{2484542}{5074091} a^{3} + \frac{2482851}{5074091} a$, $\frac{1}{98820431776769202594936970481653826703081509} a^{30} + \frac{8547186992829204746492630370049542845}{98820431776769202594936970481653826703081509} a^{28} - \frac{6064884370041987805941922285825495643}{98820431776769202594936970481653826703081509} a^{26} - \frac{3990832091229929646211043011467404898}{98820431776769202594936970481653826703081509} a^{24} + \frac{44505742807031099993322519711226879045544147}{98820431776769202594936970481653826703081509} a^{22} + \frac{18351494195360900864786766727447542415509996}{98820431776769202594936970481653826703081509} a^{20} - \frac{32255546484636259927915424002063441993579219}{98820431776769202594936970481653826703081509} a^{18} - \frac{36259157870759714436610603541421877096522394}{98820431776769202594936970481653826703081509} a^{16} - \frac{31987050077359177905283450736498183067817980}{98820431776769202594936970481653826703081509} a^{14} - \frac{14164749434341713849865317082645499428479962}{98820431776769202594936970481653826703081509} a^{12} - \frac{1512434770099872916998141477668608309524608}{8983675616069927508630633680150347882098319} a^{10} + \frac{4085276404776425843628815309815498322901108}{98820431776769202594936970481653826703081509} a^{8} + \frac{17879379883348722657725772654733163648908333}{98820431776769202594936970481653826703081509} a^{6} + \frac{13816178718835191644037266393540620815596328}{98820431776769202594936970481653826703081509} a^{4} - \frac{40117997260163266267990148239512616655050968}{98820431776769202594936970481653826703081509} a^{2} + \frac{232480135186105756443113278392785762}{2898476586977754849462371033289301829}$, $\frac{1}{98820431776769202594936970481653826703081509} a^{31} + \frac{8547186992829204746492630370049542845}{98820431776769202594936970481653826703081509} a^{29} - \frac{6064884370041987805941922285825495643}{98820431776769202594936970481653826703081509} a^{27} - \frac{3990832091229929646211043011467404898}{98820431776769202594936970481653826703081509} a^{25} - \frac{4915794253880795285843427452614742622383}{98820431776769202594936970481653826703081509} a^{23} + \frac{32635523039271366182499868814537083387834160}{98820431776769202594936970481653826703081509} a^{21} - \frac{27415223914306410306036339645537839247033353}{98820431776769202594936970481653826703081509} a^{19} + \frac{28323607613081915483286549970821829479767902}{98820431776769202594936970481653826703081509} a^{17} + \frac{40194683357804488658752544999801452295881735}{98820431776769202594936970481653826703081509} a^{15} - \frac{24302348104368217078905776976207863222819031}{98820431776769202594936970481653826703081509} a^{13} - \frac{2240175576128486671336605209593786344844441}{8983675616069927508630633680150347882098319} a^{11} + \frac{12411985162127076242106121265796605145630360}{98820431776769202594936970481653826703081509} a^{9} + \frac{19808739229564117262251002083558054254174867}{98820431776769202594936970481653826703081509} a^{7} + \frac{26593953336314340121375408663389677017142233}{98820431776769202594936970481653826703081509} a^{5} + \frac{11077722233710143194192474762200489318030307}{98820431776769202594936970481653826703081509} a^{3} - \frac{6891612857187018789698569528205876941016}{16924204791363110566010784463376233379531} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.72000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.18000.1, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.2304000.2, 4.0.2304000.1, 8.0.82944000000.6, 8.8.2621440000.1, 8.0.5308416000000.7, 8.8.33554432000000.1, 8.8.33554432000000.2, 8.0.173946175488.1, 8.0.108716359680000.13, 16.0.450868486864896000000000000.13, 16.16.1125899906842624000000000000.1, 16.0.11819246862071129702400000000.1

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

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
5Data not computed