Properties

Label 42.2.15523175110...8125.1
Degree $42$
Signature $[2, 20]$
Discriminant $3^{42}\cdot 5^{41}\cdot 7^{42}$
Root discriminant $101.05$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times S_3\times F_7$ (as 42T95)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 5)
 
gp: K = bnfinit(x^42 - 5, 1)
 
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, 1]);
 

Normalized defining polynomial

\( x^{42} - 5 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1552317511023484497006461990693717931517755755807346997289641876704990863800048828125=3^{42}\cdot 5^{41}\cdot 7^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $101.05$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  Not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2\times S_3\times F_7$ (as 42T95):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 504
The 42 conjugacy class representatives for $C_2\times S_3\times F_7$
Character table for $C_2\times S_3\times F_7$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.675.1, 6.2.2278125.1, 7.1.12867859375.2, 14.2.827909024473876953125.1, 21.1.557192518080328228128540460681915283203125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{20}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $42$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
7Data not computed