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Properties

Label 20.10.7372897256...3744.4
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{28}\cdot 13^{14}\cdot 17^{8}$
Root discriminant $49.36$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -48, 0, 21, 0, 260, 0, -94, 0, -368, 0, 66, 0, 180, 0, -9, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 24*x^18 - 9*x^16 + 180*x^14 + 66*x^12 - 368*x^10 - 94*x^8 + 260*x^6 + 21*x^4 - 48*x^2 - 1)
 
gp: K = bnfinit(x^20 - 24*x^18 - 9*x^16 + 180*x^14 + 66*x^12 - 368*x^10 - 94*x^8 + 260*x^6 + 21*x^4 - 48*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 24 x^{18} - 9 x^{16} + 180 x^{14} + 66 x^{12} - 368 x^{10} - 94 x^{8} + 260 x^{6} + 21 x^{4} - 48 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7372897256329164687850412645023744=-\,2^{28}\cdot 13^{14}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.36$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{136} a^{16} - \frac{3}{136} a^{14} + \frac{5}{136} a^{12} + \frac{3}{136} a^{10} - \frac{1}{4} a^{9} - \frac{13}{136} a^{8} + \frac{15}{136} a^{6} + \frac{19}{136} a^{4} - \frac{39}{136} a^{2} + \frac{1}{4} a + \frac{9}{34}$, $\frac{1}{136} a^{17} - \frac{3}{136} a^{15} + \frac{5}{136} a^{13} + \frac{3}{136} a^{11} - \frac{13}{136} a^{9} - \frac{1}{4} a^{8} + \frac{15}{136} a^{7} + \frac{19}{136} a^{5} - \frac{39}{136} a^{3} + \frac{9}{34} a + \frac{1}{4}$, $\frac{1}{32776} a^{18} + \frac{69}{32776} a^{16} + \frac{1829}{32776} a^{14} - \frac{351}{32776} a^{12} - \frac{2211}{32776} a^{10} - \frac{1}{4} a^{9} + \frac{5607}{32776} a^{8} + \frac{3207}{32776} a^{6} - \frac{11897}{32776} a^{4} + \frac{7607}{16388} a^{2} + \frac{1}{4} a - \frac{2123}{8194}$, $\frac{1}{65552} a^{19} - \frac{1}{65552} a^{18} + \frac{69}{65552} a^{17} - \frac{69}{65552} a^{16} - \frac{567}{16388} a^{15} + \frac{567}{16388} a^{14} - \frac{278}{4097} a^{13} + \frac{278}{4097} a^{12} + \frac{943}{32776} a^{11} - \frac{943}{32776} a^{10} - \frac{7439}{32776} a^{9} + \frac{7439}{32776} a^{8} - \frac{2271}{16388} a^{7} + \frac{2271}{16388} a^{6} - \frac{975}{8194} a^{5} - \frac{1561}{4097} a^{4} - \frac{5271}{65552} a^{3} + \frac{5271}{65552} a^{2} - \frac{20783}{65552} a - \frac{11993}{65552}$

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:  $14$
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:  \( 6861324991.65 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$