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Properties

Label 32.32.1847055309...5616.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{128}\cdot 13^{24}$
Root discriminant $109.54$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1, 0, -160256, 0, 9153840, 0, -78474144, 0, 280891560, 0, -548056352, 0, 659656088, 0, -523451648, 0, 283997723, 0, -107218880, 0, 28278912, 0, -5172960, 0, 644748, 0, -53120, 0, 2744, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 80*x^30 + 2744*x^28 - 53120*x^26 + 644748*x^24 - 5172960*x^22 + 28278912*x^20 - 107218880*x^18 + 283997723*x^16 - 523451648*x^14 + 659656088*x^12 - 548056352*x^10 + 280891560*x^8 - 78474144*x^6 + 9153840*x^4 - 160256*x^2 + 1)
 
gp: K = bnfinit(x^32 - 80*x^30 + 2744*x^28 - 53120*x^26 + 644748*x^24 - 5172960*x^22 + 28278912*x^20 - 107218880*x^18 + 283997723*x^16 - 523451648*x^14 + 659656088*x^12 - 548056352*x^10 + 280891560*x^8 - 78474144*x^6 + 9153840*x^4 - 160256*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 80 x^{30} + 2744 x^{28} - 53120 x^{26} + 644748 x^{24} - 5172960 x^{22} + 28278912 x^{20} - 107218880 x^{18} + 283997723 x^{16} - 523451648 x^{14} + 659656088 x^{12} - 548056352 x^{10} + 280891560 x^{8} - 78474144 x^{6} + 9153840 x^{4} - 160256 x^{2} + 1 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(184705530909499611281860495545984324846677532007655461458716655616=2^{128}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.54$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(416=2^{5}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(261,·)$, $\chi_{416}(129,·)$, $\chi_{416}(395,·)$, $\chi_{416}(151,·)$, $\chi_{416}(25,·)$, $\chi_{416}(411,·)$, $\chi_{416}(285,·)$, $\chi_{416}(389,·)$, $\chi_{416}(291,·)$, $\chi_{416}(135,·)$, $\chi_{416}(47,·)$, $\chi_{416}(99,·)$, $\chi_{416}(307,·)$, $\chi_{416}(53,·)$, $\chi_{416}(313,·)$, $\chi_{416}(31,·)$, $\chi_{416}(181,·)$, $\chi_{416}(203,·)$, $\chi_{416}(77,·)$, $\chi_{416}(209,·)$, $\chi_{416}(83,·)$, $\chi_{416}(343,·)$, $\chi_{416}(187,·)$, $\chi_{416}(337,·)$, $\chi_{416}(233,·)$, $\chi_{416}(359,·)$, $\chi_{416}(365,·)$, $\chi_{416}(239,·)$, $\chi_{416}(157,·)$, $\chi_{416}(105,·)$, $\chi_{416}(255,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{69} a^{24} - \frac{13}{69} a^{22} + \frac{26}{69} a^{20} - \frac{26}{69} a^{18} - \frac{10}{69} a^{16} - \frac{2}{23} a^{14} + \frac{9}{23} a^{12} - \frac{2}{23} a^{10} - \frac{2}{69} a^{8} - \frac{4}{69} a^{6} + \frac{5}{69} a^{4} + \frac{10}{69} a^{2} - \frac{10}{69}$, $\frac{1}{69} a^{25} - \frac{13}{69} a^{23} + \frac{26}{69} a^{21} - \frac{26}{69} a^{19} - \frac{10}{69} a^{17} - \frac{2}{23} a^{15} + \frac{9}{23} a^{13} - \frac{2}{23} a^{11} - \frac{2}{69} a^{9} - \frac{4}{69} a^{7} + \frac{5}{69} a^{5} + \frac{10}{69} a^{3} - \frac{10}{69} a$, $\frac{1}{69} a^{26} - \frac{5}{69} a^{22} - \frac{11}{23} a^{20} - \frac{1}{23} a^{18} + \frac{2}{69} a^{16} + \frac{6}{23} a^{14} - \frac{11}{69} a^{10} - \frac{10}{23} a^{8} + \frac{22}{69} a^{6} + \frac{2}{23} a^{4} - \frac{6}{23} a^{2} + \frac{8}{69}$, $\frac{1}{69} a^{27} - \frac{5}{69} a^{23} - \frac{11}{23} a^{21} - \frac{1}{23} a^{19} + \frac{2}{69} a^{17} + \frac{6}{23} a^{15} - \frac{11}{69} a^{11} - \frac{10}{23} a^{9} + \frac{22}{69} a^{7} + \frac{2}{23} a^{5} - \frac{6}{23} a^{3} + \frac{8}{69} a$, $\frac{1}{16833723} a^{28} + \frac{7365}{5611241} a^{26} - \frac{33686}{16833723} a^{24} - \frac{107422}{330073} a^{22} - \frac{1210642}{5611241} a^{20} - \frac{1807333}{16833723} a^{18} - \frac{62117}{243967} a^{16} - \frac{2524970}{5611241} a^{14} + \frac{2620}{132549} a^{12} - \frac{2234382}{5611241} a^{10} - \frac{6980519}{16833723} a^{8} + \frac{1533787}{5611241} a^{6} + \frac{1429850}{5611241} a^{4} + \frac{6315632}{16833723} a^{2} + \frac{538914}{5611241}$, $\frac{1}{16833723} a^{29} + \frac{7365}{5611241} a^{27} - \frac{33686}{16833723} a^{25} - \frac{107422}{330073} a^{23} - \frac{1210642}{5611241} a^{21} - \frac{1807333}{16833723} a^{19} - \frac{62117}{243967} a^{17} - \frac{2524970}{5611241} a^{15} + \frac{2620}{132549} a^{13} - \frac{2234382}{5611241} a^{11} - \frac{6980519}{16833723} a^{9} + \frac{1533787}{5611241} a^{7} + \frac{1429850}{5611241} a^{5} + \frac{6315632}{16833723} a^{3} + \frac{538914}{5611241} a$, $\frac{1}{31626682583370470426178280347} a^{30} + \frac{588385733212756664104}{31626682583370470426178280347} a^{28} - \frac{4707839059541197162751907}{10542227527790156808726093449} a^{26} - \frac{177118216990599220013870923}{31626682583370470426178280347} a^{24} - \frac{1058430518074083961220257367}{31626682583370470426178280347} a^{22} - \frac{2257021246579339047762759014}{31626682583370470426178280347} a^{20} - \frac{3309098614387289213300666991}{10542227527790156808726093449} a^{18} + \frac{528343277662332159542528728}{10542227527790156808726093449} a^{16} + \frac{11138562637146541383961741036}{31626682583370470426178280347} a^{14} - \frac{15485461176940138275779430623}{31626682583370470426178280347} a^{12} + \frac{483728415539866272838107012}{10542227527790156808726093449} a^{10} - \frac{3372368629505165775274822474}{31626682583370470426178280347} a^{8} - \frac{6972901853652496384006991708}{31626682583370470426178280347} a^{6} - \frac{3363699504020606263474473209}{31626682583370470426178280347} a^{4} + \frac{122794872054446801656325191}{10542227527790156808726093449} a^{2} + \frac{3924797260107352215359943435}{10542227527790156808726093449}$, $\frac{1}{31626682583370470426178280347} a^{31} + \frac{588385733212756664104}{31626682583370470426178280347} a^{29} - \frac{4707839059541197162751907}{10542227527790156808726093449} a^{27} - \frac{177118216990599220013870923}{31626682583370470426178280347} a^{25} - \frac{1058430518074083961220257367}{31626682583370470426178280347} a^{23} - \frac{2257021246579339047762759014}{31626682583370470426178280347} a^{21} - \frac{3309098614387289213300666991}{10542227527790156808726093449} a^{19} + \frac{528343277662332159542528728}{10542227527790156808726093449} a^{17} + \frac{11138562637146541383961741036}{31626682583370470426178280347} a^{15} - \frac{15485461176940138275779430623}{31626682583370470426178280347} a^{13} + \frac{483728415539866272838107012}{10542227527790156808726093449} a^{11} - \frac{3372368629505165775274822474}{31626682583370470426178280347} a^{9} - \frac{6972901853652496384006991708}{31626682583370470426178280347} a^{7} - \frac{3363699504020606263474473209}{31626682583370470426178280347} a^{5} + \frac{122794872054446801656325191}{10542227527790156808726093449} a^{3} + \frac{3924797260107352215359943435}{10542227527790156808726093449} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $31$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29157260098603590000000 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.4.140608.1, \(\Q(\sqrt{2}, \sqrt{13})\), 4.4.35152.1, 4.4.346112.1, \(\Q(\zeta_{16})^+\), 4.4.4499456.1, 4.4.4499456.2, 8.8.316329754624.1, 8.8.119793516544.1, 8.8.20245104295936.1, 8.8.10365493399519232.1, 8.8.10365493399519232.2, \(\Q(\zeta_{32})^+\), 8.8.61334280470528.1, 16.16.6557827967253220516257857536.1, 16.16.107443453415476764938368737869824.1, 16.16.3761893960837392421076598784.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
13Data not computed