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Properties

Label 20.20.4510514237...9296.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{24}\cdot 7^{12}\cdot 13^{8}\cdot 47^{8}$
Root discriminant $96.10$
Ramified primes $2, 7, 13, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_6$ (as 20T89)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15376, 0, -583424, 0, 1987084, 0, -2767108, 0, 1994469, 0, -811774, 0, 191379, 0, -25784, 0, 1899, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 70*x^18 + 1899*x^16 - 25784*x^14 + 191379*x^12 - 811774*x^10 + 1994469*x^8 - 2767108*x^6 + 1987084*x^4 - 583424*x^2 + 15376)
 
gp: K = bnfinit(x^20 - 70*x^18 + 1899*x^16 - 25784*x^14 + 191379*x^12 - 811774*x^10 + 1994469*x^8 - 2767108*x^6 + 1987084*x^4 - 583424*x^2 + 15376, 1)
 

Normalized defining polynomial

\( x^{20} - 70 x^{18} + 1899 x^{16} - 25784 x^{14} + 191379 x^{12} - 811774 x^{10} + 1994469 x^{8} - 2767108 x^{6} + 1987084 x^{4} - 583424 x^{2} + 15376 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4510514237834618843771703646681663799296=2^{24}\cdot 7^{12}\cdot 13^{8}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.10$
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:  $2, 7, 13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{1}{32} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{416} a^{14} - \frac{1}{32} a^{12} - \frac{3}{416} a^{10} - \frac{1}{16} a^{9} + \frac{1}{416} a^{8} - \frac{1}{8} a^{7} - \frac{47}{208} a^{6} + \frac{1}{16} a^{5} + \frac{5}{104} a^{4} + \frac{3}{8} a^{3} + \frac{21}{52} a^{2} + \frac{1}{4} a - \frac{4}{13}$, $\frac{1}{832} a^{15} - \frac{1}{832} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{23}{832} a^{11} - \frac{23}{832} a^{10} + \frac{1}{832} a^{9} - \frac{1}{832} a^{8} + \frac{11}{104} a^{7} - \frac{11}{104} a^{6} + \frac{31}{208} a^{5} - \frac{31}{208} a^{4} + \frac{17}{52} a^{3} - \frac{17}{52} a^{2} + \frac{5}{52} a - \frac{5}{52}$, $\frac{1}{78208} a^{16} - \frac{3}{39104} a^{14} + \frac{581}{19552} a^{12} + \frac{393}{39104} a^{10} - \frac{1}{16} a^{9} + \frac{407}{78208} a^{8} - \frac{1}{8} a^{7} + \frac{4085}{19552} a^{6} - \frac{3}{16} a^{5} - \frac{807}{19552} a^{4} + \frac{1}{8} a^{3} - \frac{913}{2444} a^{2} - \frac{1}{4} a + \frac{1751}{4888}$, $\frac{1}{78208} a^{17} - \frac{3}{39104} a^{15} - \frac{15}{9776} a^{13} - \frac{829}{39104} a^{11} + \frac{2851}{78208} a^{9} - \frac{6}{611} a^{7} + \frac{4081}{19552} a^{5} - \frac{151}{1222} a^{3} + \frac{529}{4888} a - \frac{1}{2}$, $\frac{1}{696460540672} a^{18} - \frac{1923843}{696460540672} a^{16} - \frac{363034259}{348230270336} a^{14} + \frac{3383978779}{348230270336} a^{12} + \frac{11447550901}{696460540672} a^{10} - \frac{6736153839}{696460540672} a^{8} + \frac{802801309}{10882195948} a^{6} - \frac{1}{4} a^{5} + \frac{42298931371}{174115135168} a^{4} - \frac{1}{4} a^{3} + \frac{16624388111}{43528783792} a^{2} - \frac{1}{2} a - \frac{6620947595}{43528783792}$, $\frac{1}{43180553521664} a^{19} - \frac{1}{1392921081344} a^{18} + \frac{131654667}{43180553521664} a^{17} - \frac{6981391}{1392921081344} a^{16} - \frac{9971781745}{21590276760832} a^{15} - \frac{447342035}{696460540672} a^{14} - \frac{331096610261}{21590276760832} a^{13} - \frac{2849664739}{696460540672} a^{12} + \frac{1042264007337}{43180553521664} a^{11} - \frac{13424512849}{1392921081344} a^{10} + \frac{1857423195083}{43180553521664} a^{9} + \frac{1437539609}{1392921081344} a^{8} - \frac{592574796687}{5397569190208} a^{7} + \frac{38589094781}{174115135168} a^{6} - \frac{1876966005543}{10795138380416} a^{5} + \frac{43574240091}{348230270336} a^{4} + \frac{3747419747}{2698784595104} a^{3} + \frac{25586421049}{87057567584} a^{2} + \frac{831655444527}{2698784595104} a - \frac{39107428995}{87057567584}$

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:  $19$
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:  \( 291227711962000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 20T89):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

10.10.67160362103212478464.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 10 sibling: data not computed
Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47Data not computed