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Properties

Label 20.14.4414566420...6624.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{10}\cdot 401^{11}$
Root discriminant $38.22$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-401, 0, -2406, 0, 32794, 0, -57207, 0, 33596, 0, -3375, 0, -2899, 0, 708, 0, 13, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 13*x^16 + 708*x^14 - 2899*x^12 - 3375*x^10 + 33596*x^8 - 57207*x^6 + 32794*x^4 - 2406*x^2 - 401)
 
gp: K = bnfinit(x^20 - 15*x^18 + 13*x^16 + 708*x^14 - 2899*x^12 - 3375*x^10 + 33596*x^8 - 57207*x^6 + 32794*x^4 - 2406*x^2 - 401, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 13 x^{16} + 708 x^{14} - 2899 x^{12} - 3375 x^{10} + 33596 x^{8} - 57207 x^{6} + 32794 x^{4} - 2406 x^{2} - 401 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44145664201696349530694455706624=-\,2^{10}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{8466} a^{16} - \frac{77}{498} a^{14} - \frac{1}{6} a^{13} + \frac{433}{2822} a^{12} + \frac{677}{8466} a^{10} + \frac{1}{3} a^{9} - \frac{169}{1411} a^{8} - \frac{1975}{8466} a^{6} - \frac{1}{6} a^{5} + \frac{262}{1411} a^{4} - \frac{1}{2} a^{3} + \frac{41}{249} a^{2} - \frac{1}{6} a - \frac{1511}{4233}$, $\frac{1}{8466} a^{17} + \frac{1}{83} a^{15} - \frac{1}{6} a^{14} - \frac{56}{4233} a^{13} - \frac{1}{6} a^{12} + \frac{348}{1411} a^{11} + \frac{1}{3} a^{10} - \frac{2425}{8466} a^{9} + \frac{1}{3} a^{8} + \frac{1223}{2822} a^{7} - \frac{1}{6} a^{6} + \frac{161}{8466} a^{5} + \frac{1}{3} a^{4} - \frac{14}{83} a^{3} + \frac{1}{3} a^{2} - \frac{100}{4233} a - \frac{1}{6}$, $\frac{1}{77638309429026} a^{18} + \frac{33698921}{12939718238171} a^{16} - \frac{463646372455}{12939718238171} a^{14} - \frac{1969845758273}{77638309429026} a^{12} - \frac{1}{2} a^{11} + \frac{900003372965}{25879436476342} a^{10} - \frac{1}{2} a^{9} + \frac{36798520016377}{77638309429026} a^{8} - \frac{1}{2} a^{7} - \frac{468389767715}{1522319792726} a^{6} - \frac{1}{2} a^{5} - \frac{14198697082895}{77638309429026} a^{4} - \frac{6914350366495}{77638309429026} a^{2} - \frac{1}{2} a - \frac{24601872925229}{77638309429026}$, $\frac{1}{77638309429026} a^{19} + \frac{33698921}{12939718238171} a^{17} - \frac{463646372455}{12939718238171} a^{15} - \frac{1969845758273}{77638309429026} a^{13} - \frac{1}{6} a^{12} + \frac{900003372965}{25879436476342} a^{11} - \frac{1}{2} a^{10} + \frac{36798520016377}{77638309429026} a^{9} - \frac{1}{6} a^{8} - \frac{468389767715}{1522319792726} a^{7} - \frac{1}{2} a^{6} - \frac{14198697082895}{77638309429026} a^{5} + \frac{1}{3} a^{4} - \frac{6914350366495}{77638309429026} a^{3} - \frac{1}{2} a^{2} - \frac{24601872925229}{77638309429026} a + \frac{1}{3}$

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:  $16$
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:  \( 618100118.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.8$x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
401Data not computed