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Properties

Label 20.8.19103464373...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}$
Root discriminant $29.11$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 10, 0, -91, 0, -236, 0, 258, 0, 576, 0, -38, 0, -196, 0, -35, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 - 35*x^16 - 196*x^14 - 38*x^12 + 576*x^10 + 258*x^8 - 236*x^6 - 91*x^4 + 10*x^2 + 1)
 
gp: K = bnfinit(x^20 + 6*x^18 - 35*x^16 - 196*x^14 - 38*x^12 + 576*x^10 + 258*x^8 - 236*x^6 - 91*x^4 + 10*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 6 x^{18} - 35 x^{16} - 196 x^{14} - 38 x^{12} + 576 x^{10} + 258 x^{8} - 236 x^{6} - 91 x^{4} + 10 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(191034643739330160640000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.11$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{16} a - \frac{5}{16}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{4} + \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{16}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{9}{32} a + \frac{7}{32}$, $\frac{1}{774848} a^{18} - \frac{9973}{774848} a^{16} + \frac{31}{24214} a^{14} - \frac{5013}{193712} a^{12} - \frac{2813}{387424} a^{10} - \frac{17025}{387424} a^{8} + \frac{6951}{48428} a^{6} - \frac{12163}{193712} a^{4} - \frac{1}{2} a^{3} + \frac{249657}{774848} a^{2} - \frac{1}{2} a + \frac{196551}{774848}$, $\frac{1}{1549696} a^{19} - \frac{1}{1549696} a^{18} - \frac{9973}{1549696} a^{17} + \frac{9973}{1549696} a^{16} - \frac{11859}{387424} a^{15} + \frac{11859}{387424} a^{14} + \frac{3547}{193712} a^{13} - \frac{3547}{193712} a^{12} - \frac{27027}{774848} a^{11} + \frac{27027}{774848} a^{10} + \frac{7189}{774848} a^{9} - \frac{7189}{774848} a^{8} - \frac{8517}{387424} a^{7} - \frac{88339}{387424} a^{6} + \frac{12079}{193712} a^{5} + \frac{36349}{193712} a^{4} + \frac{491797}{1549696} a^{3} + \frac{670475}{1549696} a^{2} - \frac{45589}{1549696} a + \frac{433013}{1549696}$

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:  $13$
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:  \( 14571316.1474 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$