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Properties

Label 20.0.14980811816...1600.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{2}\cdot 29^{4}\cdot 53^{4}$
Root discriminant $14.41$
Ramified primes $2, 5, 29, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 10, 0, -37, 0, -58, 0, 75, 0, 24, 0, -48, 0, 6, 0, 13, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 + 13*x^16 + 6*x^14 - 48*x^12 + 24*x^10 + 75*x^8 - 58*x^6 - 37*x^4 + 10*x^2 + 25)
 
gp: K = bnfinit(x^20 - 6*x^18 + 13*x^16 + 6*x^14 - 48*x^12 + 24*x^10 + 75*x^8 - 58*x^6 - 37*x^4 + 10*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} + 13 x^{16} + 6 x^{14} - 48 x^{12} + 24 x^{10} + 75 x^{8} - 58 x^{6} - 37 x^{4} + 10 x^{2} + 25 \)

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:  \(149808118161024326041600=2^{30}\cdot 5^{2}\cdot 29^{4}\cdot 53^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.41$
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:  $2, 5, 29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{65904380} a^{18} + \frac{5682179}{65904380} a^{16} + \frac{5213759}{32952190} a^{14} - \frac{1}{4} a^{13} + \frac{669261}{65904380} a^{12} - \frac{15623903}{65904380} a^{10} - \frac{1}{4} a^{9} + \frac{4755139}{65904380} a^{8} - \frac{1}{4} a^{7} + \frac{146491}{3295219} a^{6} - \frac{1}{2} a^{5} - \frac{7401362}{16476095} a^{4} + \frac{7980879}{32952190} a^{2} + \frac{1}{4} a + \frac{2117305}{6590438}$, $\frac{1}{65904380} a^{19} + \frac{5682179}{65904380} a^{17} - \frac{6048577}{65904380} a^{15} - \frac{7903417}{32952190} a^{13} + \frac{213048}{16476095} a^{11} - \frac{1}{4} a^{10} + \frac{4755139}{65904380} a^{9} - \frac{1}{4} a^{8} + \frac{3881183}{13180876} a^{7} - \frac{1}{2} a^{6} - \frac{7401362}{16476095} a^{5} + \frac{1}{4} a^{4} - \frac{514337}{65904380} a^{3} - \frac{1}{4} a^{2} + \frac{939391}{13180876} a - \frac{1}{4}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{981}{16942} a^{18} + \frac{5357}{16942} a^{16} - \frac{10391}{16942} a^{14} - \frac{8657}{16942} a^{12} + \frac{18261}{8471} a^{10} - \frac{3246}{8471} a^{8} - \frac{30857}{8471} a^{6} + \frac{9668}{8471} a^{4} + \frac{14895}{16942} a^{2} + \frac{5678}{8471} \) (order $4$)
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:  \( 4718.99191561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 65 conjugacy class representatives for t20n638 are not computed
Character table for t20n638 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.2419065856.1

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 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
53Data not computed