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Properties

Label 32.0.16558545548...8336.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 17^{16}$
Root discriminant $65.97$
Ramified primes $2, 17$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4294967296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3437249, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 3437249*x^16 + 4294967296)
 
gp: K = bnfinit(x^32 + 3437249*x^16 + 4294967296, 1)
 

Normalized defining polynomial

\( x^{32} + 3437249 x^{16} + 4294967296 \)

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:  \(16558545548645863110798599687464368531777983698317749518336=2^{128}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.97$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(544=2^{5}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{544}(1,·)$, $\chi_{544}(135,·)$, $\chi_{544}(137,·)$, $\chi_{544}(271,·)$, $\chi_{544}(273,·)$, $\chi_{544}(407,·)$, $\chi_{544}(409,·)$, $\chi_{544}(543,·)$, $\chi_{544}(33,·)$, $\chi_{544}(35,·)$, $\chi_{544}(169,·)$, $\chi_{544}(171,·)$, $\chi_{544}(305,·)$, $\chi_{544}(307,·)$, $\chi_{544}(441,·)$, $\chi_{544}(443,·)$, $\chi_{544}(67,·)$, $\chi_{544}(69,·)$, $\chi_{544}(203,·)$, $\chi_{544}(205,·)$, $\chi_{544}(339,·)$, $\chi_{544}(341,·)$, $\chi_{544}(475,·)$, $\chi_{544}(477,·)$, $\chi_{544}(101,·)$, $\chi_{544}(103,·)$, $\chi_{544}(237,·)$, $\chi_{544}(239,·)$, $\chi_{544}(373,·)$, $\chi_{544}(375,·)$, $\chi_{544}(509,·)$, $\chi_{544}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{833049} a^{16} - \frac{363998}{833049}$, $\frac{1}{3332196} a^{17} - \frac{1197047}{3332196} a$, $\frac{1}{13328784} a^{18} + \frac{5467345}{13328784} a^{2}$, $\frac{1}{53315136} a^{19} - \frac{7861439}{53315136} a^{3}$, $\frac{1}{213260544} a^{20} + \frac{98768833}{213260544} a^{4}$, $\frac{1}{853042176} a^{21} - \frac{114491711}{853042176} a^{5}$, $\frac{1}{3412168704} a^{22} + \frac{738550465}{3412168704} a^{6}$, $\frac{1}{13648674816} a^{23} + \frac{4150719169}{13648674816} a^{7}$, $\frac{1}{54594699264} a^{24} - \frac{23146630463}{54594699264} a^{8}$, $\frac{1}{218378797056} a^{25} + \frac{86042768065}{218378797056} a^{9}$, $\frac{1}{873515188224} a^{26} - \frac{350714826047}{873515188224} a^{10}$, $\frac{1}{3494060752896} a^{27} - \frac{1224230014271}{3494060752896} a^{11}$, $\frac{1}{13976243011584} a^{28} - \frac{1224230014271}{13976243011584} a^{12}$, $\frac{1}{55904972046336} a^{29} + \frac{26728256008897}{55904972046336} a^{13}$, $\frac{1}{223619888185344} a^{30} - \frac{85081688083775}{223619888185344} a^{14}$, $\frac{1}{894479552741376} a^{31} + \frac{362158088286913}{894479552741376} a^{15}$

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:  \( \frac{7589}{3494060752896} a^{27} + \frac{25963647845}{3494060752896} a^{11} \) (order $32$)
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

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-34}) \), \(\Q(i, \sqrt{17})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.4.591872.2, 4.0.591872.5, 8.0.5473632256.1, \(\Q(\zeta_{16})\), 8.0.1401249857536.3, 8.8.350312464384.1, 8.0.350312464384.1, 8.0.1401249857536.2, 8.0.1401249857536.1, 8.0.2147483648.1, \(\Q(\zeta_{32})^+\), 8.0.179359981764608.32, 8.8.179359981764608.1, 16.0.1963501163244660295991296.1, \(\Q(\zeta_{32})\), 16.0.128680012234402057158085574656.5, 16.0.32170003058600514289521393664.1, 16.16.32170003058600514289521393664.1, 16.0.128680012234402057158085574656.3, 16.0.128680012234402057158085574656.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$