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Properties

Label 20.0.29841354420...9392.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 13^{15}$
Root discriminant $118.51$
Ramified primes $2, 11, 13$
Class number $1726100$ (GRH)
Class group $[2, 863050]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44926453, 0, 224632265, 0, 352499862, 0, 259191075, 0, 103144756, 0, 23369489, 0, 3007862, 0, 211926, 0, 7865, 0, 143, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 143*x^18 + 7865*x^16 + 211926*x^14 + 3007862*x^12 + 23369489*x^10 + 103144756*x^8 + 259191075*x^6 + 352499862*x^4 + 224632265*x^2 + 44926453)
 
gp: K = bnfinit(x^20 + 143*x^18 + 7865*x^16 + 211926*x^14 + 3007862*x^12 + 23369489*x^10 + 103144756*x^8 + 259191075*x^6 + 352499862*x^4 + 224632265*x^2 + 44926453, 1)
 

Normalized defining polynomial

\( x^{20} + 143 x^{18} + 7865 x^{16} + 211926 x^{14} + 3007862 x^{12} + 23369489 x^{10} + 103144756 x^{8} + 259191075 x^{6} + 352499862 x^{4} + 224632265 x^{2} + 44926453 \)

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:  \(298413544204512013291051431427069566779392=2^{20}\cdot 11^{18}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.51$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(572=2^{2}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{572}(1,·)$, $\chi_{572}(389,·)$, $\chi_{572}(521,·)$, $\chi_{572}(395,·)$, $\chi_{572}(83,·)$, $\chi_{572}(343,·)$, $\chi_{572}(25,·)$, $\chi_{572}(157,·)$, $\chi_{572}(359,·)$, $\chi_{572}(493,·)$, $\chi_{572}(239,·)$, $\chi_{572}(53,·)$, $\chi_{572}(307,·)$, $\chi_{572}(181,·)$, $\chi_{572}(567,·)$, $\chi_{572}(441,·)$, $\chi_{572}(447,·)$, $\chi_{572}(151,·)$, $\chi_{572}(313,·)$, $\chi_{572}(255,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{169} a^{8}$, $\frac{1}{169} a^{9}$, $\frac{1}{5577} a^{10} - \frac{1}{507} a^{8} + \frac{1}{39} a^{6} - \frac{1}{39} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{5577} a^{11} - \frac{1}{507} a^{9} + \frac{1}{39} a^{7} - \frac{1}{39} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{72501} a^{12} - \frac{1}{507} a^{8} + \frac{1}{39} a^{6} + \frac{1}{3}$, $\frac{1}{72501} a^{13} - \frac{1}{507} a^{9} + \frac{1}{39} a^{7} + \frac{1}{3} a$, $\frac{1}{72501} a^{14} - \frac{1}{507} a^{8} - \frac{1}{39} a^{6} + \frac{1}{39} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{72501} a^{15} - \frac{1}{507} a^{9} - \frac{1}{39} a^{7} + \frac{1}{39} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{21677799} a^{16} - \frac{8}{1667523} a^{14} - \frac{1}{151593} a^{12} + \frac{4}{128271} a^{10} - \frac{8}{3887} a^{8} + \frac{10}{897} a^{6} - \frac{1}{299} a^{4} + \frac{3}{23} a^{2} + \frac{5}{69}$, $\frac{1}{21677799} a^{17} - \frac{8}{1667523} a^{15} - \frac{1}{151593} a^{13} + \frac{4}{128271} a^{11} - \frac{8}{3887} a^{9} + \frac{10}{897} a^{7} - \frac{1}{299} a^{5} + \frac{3}{23} a^{3} + \frac{5}{69} a$, $\frac{1}{245154228891} a^{18} + \frac{441}{81718076297} a^{16} - \frac{29804}{18858017607} a^{14} + \frac{2421}{6286005869} a^{12} + \frac{10524}{483538913} a^{10} - \frac{78646}{131874249} a^{8} - \frac{201082}{10144173} a^{6} - \frac{121701}{3381391} a^{4} + \frac{68675}{260107} a^{2} - \frac{10667}{780321}$, $\frac{1}{245154228891} a^{19} + \frac{441}{81718076297} a^{17} - \frac{29804}{18858017607} a^{15} + \frac{2421}{6286005869} a^{13} + \frac{10524}{483538913} a^{11} - \frac{78646}{131874249} a^{9} - \frac{201082}{10144173} a^{7} - \frac{121701}{3381391} a^{5} + \frac{68675}{260107} a^{3} - \frac{10667}{780321} a$

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

Class group and class number

$C_{2}\times C_{863050}$, which has order $1726100$ (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:  \( -1 \) (order $2$)
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:  \( 2015201.7241994622 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.4253392.1, \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed
13Data not computed