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Properties

Label 32.0.22642953876...8496.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 13^{16}$
Root discriminant $57.69$
Ramified primes $2, 13$
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![43046721, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 625279, 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 + 625279*x^16 + 43046721)
 
gp: K = bnfinit(x^32 + 625279*x^16 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} + 625279 x^{16} + 43046721 \)

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:  \(226429538761357513323150294281958672053835192027364458496=2^{128}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.69$
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$$\chi_{416}(1,·)$, $\chi_{416}(259,·)$, $\chi_{416}(389,·)$, $\chi_{416}(129,·)$, $\chi_{416}(103,·)$, $\chi_{416}(131,·)$, $\chi_{416}(25,·)$, $\chi_{416}(155,·)$, $\chi_{416}(157,·)$, $\chi_{416}(415,·)$, $\chi_{416}(27,·)$, $\chi_{416}(261,·)$, $\chi_{416}(363,·)$, $\chi_{416}(391,·)$, $\chi_{416}(285,·)$, $\chi_{416}(51,·)$, $\chi_{416}(53,·)$, $\chi_{416}(311,·)$, $\chi_{416}(313,·)$, $\chi_{416}(287,·)$, $\chi_{416}(181,·)$, $\chi_{416}(183,·)$, $\chi_{416}(77,·)$, $\chi_{416}(79,·)$, $\chi_{416}(209,·)$, $\chi_{416}(339,·)$, $\chi_{416}(207,·)$, $\chi_{416}(337,·)$, $\chi_{416}(105,·)$, $\chi_{416}(235,·)$, $\chi_{416}(365,·)$, $\chi_{416}(233,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{173383} a^{16} + \frac{52565}{173383}$, $\frac{1}{520149} a^{17} - \frac{120818}{520149} a$, $\frac{1}{1560447} a^{18} - \frac{640967}{1560447} a^{2}$, $\frac{1}{4681341} a^{19} - \frac{640967}{4681341} a^{3}$, $\frac{1}{14044023} a^{20} - \frac{5322308}{14044023} a^{4}$, $\frac{1}{42132069} a^{21} - \frac{19366331}{42132069} a^{5}$, $\frac{1}{126396207} a^{22} + \frac{22765738}{126396207} a^{6}$, $\frac{1}{379188621} a^{23} + \frac{149161945}{379188621} a^{7}$, $\frac{1}{1137565863} a^{24} - \frac{230026676}{1137565863} a^{8}$, $\frac{1}{3412697589} a^{25} + \frac{907539187}{3412697589} a^{9}$, $\frac{1}{10238092767} a^{26} + \frac{4320236776}{10238092767} a^{10}$, $\frac{1}{30714278301} a^{27} - \frac{5917855991}{30714278301} a^{11}$, $\frac{1}{92142834903} a^{28} - \frac{36632134292}{92142834903} a^{12}$, $\frac{1}{276428504709} a^{29} - \frac{36632134292}{276428504709} a^{13}$, $\frac{1}{829285514127} a^{30} - \frac{36632134292}{829285514127} a^{14}$, $\frac{1}{2487856542381} a^{31} - \frac{36632134292}{2487856542381} a^{15}$

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:  \( -\frac{19}{42132069} a^{21} - \frac{11228332}{42132069} a^{5} \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(i, \sqrt{13})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{26})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.4.346112.1, 4.0.346112.2, 8.0.1871773696.1, \(\Q(\zeta_{16})\), 8.0.479174066176.3, 8.8.119793516544.1, 8.0.119793516544.1, 8.0.479174066176.2, 8.0.479174066176.1, 8.0.2147483648.1, \(\Q(\zeta_{32})^+\), 8.0.61334280470528.11, 8.8.61334280470528.1, 16.0.229607785695641627262976.1, \(\Q(\zeta_{32})\), 16.0.15047575843349569684306395136.3, 16.0.3761893960837392421076598784.1, 16.16.3761893960837392421076598784.1, 16.0.15047575843349569684306395136.1, 16.0.15047575843349569684306395136.2

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 ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
13Data not computed