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Properties

Label 46.2.12391693186...3125.1
Degree $46$
Signature $[2, 22]$
Discriminant $5^{45}\cdot 23^{46}$
Root discriminant $111.05$
Ramified primes $5, 23$
Class number Not computed
Class group Not computed
Galois group 46T6

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 5)
 
gp: K = bnfinit(x^46 - 5, 1)
 

Normalized defining polynomial

\( x^{46} - 5 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12391693186733410362886430147153084686853607607137680458329824915608696755953133106231689453125=5^{45}\cdot 23^{46}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.05$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2} a^{23} - \frac{1}{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{32} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{33} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{34} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{35} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{36} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{37} - \frac{1}{2} a^{14}$, $\frac{1}{2} a^{38} - \frac{1}{2} a^{15}$, $\frac{1}{2} a^{39} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{40} - \frac{1}{2} a^{17}$, $\frac{1}{2} a^{41} - \frac{1}{2} a^{18}$, $\frac{1}{2} a^{42} - \frac{1}{2} a^{19}$, $\frac{1}{2} a^{43} - \frac{1}{2} a^{20}$, $\frac{1}{2} a^{44} - \frac{1}{2} a^{21}$, $\frac{1}{2} a^{45} - \frac{1}{2} a^{22}$

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:  $23$
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

46T6:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1012
The 46 conjugacy class representatives for t46n6
Character table for t46n6 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 23.1.49782915114993839345824796940248012542724609375.1

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

Sibling fields

Degree 46 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ $22^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R $22^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ $22^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $22^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $22^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $22^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $22^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $46$ $22^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
23Data not computed