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Properties

Label 20.0.16468295313...2944.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $45.80$
Ramified primes $2, 7, 11$
Class number $124$ (GRH)
Class group $[124]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![282475249, 0, 40353607, 0, 5764801, 0, 823543, 0, 117649, 0, 16807, 0, 2401, 0, 343, 0, 49, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 7*x^18 + 49*x^16 + 343*x^14 + 2401*x^12 + 16807*x^10 + 117649*x^8 + 823543*x^6 + 5764801*x^4 + 40353607*x^2 + 282475249)
 
gp: K = bnfinit(x^20 + 7*x^18 + 49*x^16 + 343*x^14 + 2401*x^12 + 16807*x^10 + 117649*x^8 + 823543*x^6 + 5764801*x^4 + 40353607*x^2 + 282475249, 1)
 

Normalized defining polynomial

\( x^{20} + 7 x^{18} + 49 x^{16} + 343 x^{14} + 2401 x^{12} + 16807 x^{10} + 117649 x^{8} + 823543 x^{6} + 5764801 x^{4} + 40353607 x^{2} + 282475249 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1646829531350307068613612031442944=2^{20}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.80$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(195,·)$, $\chi_{308}(197,·)$, $\chi_{308}(139,·)$, $\chi_{308}(141,·)$, $\chi_{308}(83,·)$, $\chi_{308}(85,·)$, $\chi_{308}(279,·)$, $\chi_{308}(281,·)$, $\chi_{308}(27,·)$, $\chi_{308}(29,·)$, $\chi_{308}(223,·)$, $\chi_{308}(225,·)$, $\chi_{308}(167,·)$, $\chi_{308}(169,·)$, $\chi_{308}(111,·)$, $\chi_{308}(113,·)$, $\chi_{308}(307,·)$, $\chi_{308}(57,·)$, $\chi_{308}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$, $\frac{1}{5764801} a^{16}$, $\frac{1}{5764801} a^{17}$, $\frac{1}{40353607} a^{18}$, $\frac{1}{40353607} a^{19}$

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

Class group and class number

$C_{124}$, which has order $124$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{117649} a^{12} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8013735.51231 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{7}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, \(\Q(\zeta_{11})\), 10.10.3689195226078208.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
7Data not computed
11Data not computed