↑

Properties

Label 20.12.2829296461...9169.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{6}\cdot 19^{2}\cdot 401^{10}$
Root discriminant $37.38$
Ramified primes $3, 19, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-81, 1296, -3330, -4068, 27834, -43347, 27121, -1166, -8441, 7529, -6187, 4800, -3180, 1916, -1020, 413, -93, -6, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 14*x^18 - 6*x^17 - 93*x^16 + 413*x^15 - 1020*x^14 + 1916*x^13 - 3180*x^12 + 4800*x^11 - 6187*x^10 + 7529*x^9 - 8441*x^8 - 1166*x^7 + 27121*x^6 - 43347*x^5 + 27834*x^4 - 4068*x^3 - 3330*x^2 + 1296*x - 81)
 
gp: K = bnfinit(x^20 - 6*x^19 + 14*x^18 - 6*x^17 - 93*x^16 + 413*x^15 - 1020*x^14 + 1916*x^13 - 3180*x^12 + 4800*x^11 - 6187*x^10 + 7529*x^9 - 8441*x^8 - 1166*x^7 + 27121*x^6 - 43347*x^5 + 27834*x^4 - 4068*x^3 - 3330*x^2 + 1296*x - 81, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 14 x^{18} - 6 x^{17} - 93 x^{16} + 413 x^{15} - 1020 x^{14} + 1916 x^{13} - 3180 x^{12} + 4800 x^{11} - 6187 x^{10} + 7529 x^{9} - 8441 x^{8} - 1166 x^{7} + 27121 x^{6} - 43347 x^{5} + 27834 x^{4} - 4068 x^{3} - 3330 x^{2} + 1296 x - 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28292964615551518200697789739169=3^{6}\cdot 19^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.38$
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:  $3, 19, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{15} - \frac{1}{9} a^{14} + \frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{1}{18} a^{7} + \frac{4}{9} a^{6} + \frac{7}{18} a^{5} - \frac{1}{18} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{17} - \frac{1}{18} a^{14} + \frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} + \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{7}{18} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{18} a^{18} - \frac{1}{18} a^{15} + \frac{1}{18} a^{14} - \frac{1}{6} a^{12} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} + \frac{1}{9} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{5}{18} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6392040069763562134340286} a^{19} + \frac{3881215299975205751265}{236742224806057856827418} a^{18} - \frac{106836305763737851469035}{6392040069763562134340286} a^{17} - \frac{54345035229121451211413}{2130680023254520711446762} a^{16} - \frac{49274346630949731268835}{710226674418173570482254} a^{15} - \frac{240559348107374503095434}{3196020034881781067170143} a^{14} + \frac{171009665594251496786714}{1065340011627260355723381} a^{13} + \frac{123838943960076811609229}{6392040069763562134340286} a^{12} - \frac{901676577578986295647}{118371112403028928413709} a^{11} + \frac{79086965933490076077328}{1065340011627260355723381} a^{10} + \frac{266243218070789630122168}{3196020034881781067170143} a^{9} + \frac{997399358528400785181019}{3196020034881781067170143} a^{8} + \frac{912961712531078287932362}{3196020034881781067170143} a^{7} - \frac{516982526590589685977020}{3196020034881781067170143} a^{6} + \frac{3062439721136560730025013}{6392040069763562134340286} a^{5} + \frac{7015549624637684646673}{710226674418173570482254} a^{4} - \frac{916107006522318567448327}{2130680023254520711446762} a^{3} + \frac{23669952316494921251530}{118371112403028928413709} a^{2} + \frac{110972978918400791525374}{355113337209086785241127} a - \frac{84443784970375606260751}{236742224806057856827418}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 552558847.244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed