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Properties

Label 32.0.16595960836...9136.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 3^{16}\cdot 17^{16}$
Root discriminant $57.13$
Ramified primes $2, 3, 17$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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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, -123797504, 0, 0, 0, 0, 0, 0, 0, 3502785, 0, 0, 0, 0, 0, 0, 0, -1889, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 1889*x^24 + 3502785*x^16 - 123797504*x^8 + 4294967296)
 
gp: K = bnfinit(x^32 - 1889*x^24 + 3502785*x^16 - 123797504*x^8 + 4294967296, 1)
 

Normalized defining polynomial

\( x^{32} - 1889 x^{24} + 3502785 x^{16} - 123797504 x^{8} + 4294967296 \)

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:  \(165959608368191494870637405649052436842728987849325019136=2^{96}\cdot 3^{16}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.13$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(816=2^{4}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{816}(1,·)$, $\chi_{816}(647,·)$, $\chi_{816}(137,·)$, $\chi_{816}(781,·)$, $\chi_{816}(271,·)$, $\chi_{816}(407,·)$, $\chi_{816}(409,·)$, $\chi_{816}(545,·)$, $\chi_{816}(35,·)$, $\chi_{816}(679,·)$, $\chi_{816}(169,·)$, $\chi_{816}(815,·)$, $\chi_{816}(305,·)$, $\chi_{816}(307,·)$, $\chi_{816}(443,·)$, $\chi_{816}(577,·)$, $\chi_{816}(67,·)$, $\chi_{816}(713,·)$, $\chi_{816}(203,·)$, $\chi_{816}(205,·)$, $\chi_{816}(341,·)$, $\chi_{816}(475,·)$, $\chi_{816}(101,·)$, $\chi_{816}(611,·)$, $\chi_{816}(613,·)$, $\chi_{816}(715,·)$, $\chi_{816}(103,·)$, $\chi_{816}(749,·)$, $\chi_{816}(239,·)$, $\chi_{816}(373,·)$, $\chi_{816}(509,·)$, $\chi_{816}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{441} a^{16} + \frac{158}{441} a^{8} - \frac{173}{441}$, $\frac{1}{1764} a^{17} + \frac{599}{1764} a^{9} + \frac{709}{1764} a$, $\frac{1}{7056} a^{18} - \frac{2929}{7056} a^{10} - \frac{1055}{7056} a^{2}$, $\frac{1}{28224} a^{19} + \frac{4127}{28224} a^{11} + \frac{13057}{28224} a^{3}$, $\frac{1}{112896} a^{20} - \frac{52321}{112896} a^{12} - \frac{15167}{112896} a^{4}$, $\frac{1}{451584} a^{21} + \frac{60575}{451584} a^{13} + \frac{210625}{451584} a^{5}$, $\frac{1}{1806336} a^{22} - \frac{391009}{1806336} a^{14} - \frac{240959}{1806336} a^{6}$, $\frac{1}{7225344} a^{23} - \frac{2197345}{7225344} a^{15} + \frac{3371713}{7225344} a^{7}$, $\frac{1}{101235306332160} a^{24} + \frac{9055}{28901376} a^{16} + \frac{1638401}{28901376} a^{8} - \frac{66554804}{1544728185}$, $\frac{1}{404941225328640} a^{25} + \frac{9055}{115605504} a^{17} + \frac{30539777}{115605504} a^{9} + \frac{1478173381}{6178912740} a$, $\frac{1}{1619764901314560} a^{26} + \frac{9055}{462422016} a^{18} - \frac{200671231}{462422016} a^{10} + \frac{1478173381}{24715650960} a^{2}$, $\frac{1}{6479059605258240} a^{27} + \frac{9055}{1849688064} a^{19} - \frac{663093247}{1849688064} a^{11} - \frac{47953128539}{98862603840} a^{3}$, $\frac{1}{25916238421032960} a^{28} + \frac{9055}{7398752256} a^{20} + \frac{1186594817}{7398752256} a^{12} - \frac{47953128539}{395450415360} a^{4}$, $\frac{1}{103664953684131840} a^{29} + \frac{9055}{29595009024} a^{21} - \frac{6212157439}{29595009024} a^{13} - \frac{47953128539}{1581801661440} a^{5}$, $\frac{1}{414659814736527360} a^{30} + \frac{9055}{118380036096} a^{22} + \frac{23382851585}{118380036096} a^{14} + \frac{3115650194341}{6327206645760} a^{6}$, $\frac{1}{1658639258946109440} a^{31} + \frac{9055}{473520144384} a^{23} - \frac{94997184511}{473520144384} a^{15} + \frac{9442856840101}{25308826583040} a^{7}$

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:  \( -\frac{1}{6178912740} a^{25} - \frac{3956947021}{6178912740} a \) (order $48$)
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

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{-102}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{102})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-6}, \sqrt{-17})\), \(\Q(\sqrt{6}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{-17})\), \(\Q(\sqrt{-6}, \sqrt{17})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{51})\), \(\Q(i, \sqrt{34})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{51})\), \(\Q(\sqrt{-2}, \sqrt{-51})\), \(\Q(\sqrt{-3}, \sqrt{-34})\), \(\Q(\sqrt{3}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{-51})\), \(\Q(\sqrt{-2}, \sqrt{51})\), \(\Q(\sqrt{3}, \sqrt{-34})\), \(\Q(\sqrt{-3}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-34})\), \(\Q(\sqrt{-6}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{34})\), \(\Q(\sqrt{6}, \sqrt{-34})\), 4.4.18432.1, 4.0.18432.2, 4.4.591872.2, 4.0.591872.5, 4.0.5326848.5, 4.4.5326848.2, 4.0.2048.2, \(\Q(\zeta_{16})^+\), 8.0.443364212736.3, 8.0.443364212736.8, 8.0.443364212736.9, 8.0.5473632256.1, 8.0.1731891456.1, \(\Q(\zeta_{24})\), 8.0.443364212736.2, 8.0.443364212736.6, 8.0.443364212736.1, 8.8.443364212736.1, 8.0.27710263296.5, 8.0.443364212736.5, 8.0.443364212736.7, 8.0.27710263296.1, 8.0.443364212736.4, 8.0.1358954496.4, 8.0.1401249857536.3, 8.0.113501238460416.79, \(\Q(\zeta_{16})\), 8.8.113501238460416.6, 8.0.113501238460416.82, 8.0.113501238460416.65, 8.8.113501238460416.3, 8.0.28375309615104.64, 8.0.28375309615104.58, 8.0.28375309615104.148, 8.0.28375309615104.94, 8.0.113501238460416.96, 8.0.113501238460416.9, 8.0.1401249857536.1, 8.0.1401249857536.2, 8.8.28375309615104.8, 8.0.28375309615104.87, 8.8.350312464384.1, 8.0.350312464384.1, 8.0.339738624.1, 8.0.339738624.2, 8.0.28375309615104.155, 8.0.28375309615104.152, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, 8.8.113501238460416.4, 8.0.113501238460416.68, 16.0.196571825135013064605696.1, 16.0.12882531132048216201998893056.2, 16.0.12882531132048216201998893056.7, 16.0.12882531132048216201998893056.1, 16.0.1963501163244660295991296.1, \(\Q(\zeta_{48})\), 16.0.12882531132048216201998893056.8, 16.0.12882531132048216201998893056.6, 16.0.12882531132048216201998893056.5, 16.16.12882531132048216201998893056.1, 16.0.12882531132048216201998893056.3, 16.0.12882531132048216201998893056.4, 16.0.12882531132048216201998893056.9, 16.0.805158195753013512624930816.4, 16.0.805158195753013512624930816.3

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$