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Properties

Label 20.0.11803130064...1296.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 13^{6}\cdot 457^{6}$
Root discriminant $35.78$
Ramified primes $2, 13, 457$
Class number $42$ (GRH)
Class group $[42]$ (GRH)
Galois group $C_2\times S_5$ (as 20T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -1024, 256, -256, 384, -384, 288, -192, 128, -96, 80, -48, 32, -24, 18, -12, 6, -2, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 2*x^17 + 6*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 32*x^12 - 48*x^11 + 80*x^10 - 96*x^9 + 128*x^8 - 192*x^7 + 288*x^6 - 384*x^5 + 384*x^4 - 256*x^3 + 256*x^2 - 1024*x + 1024)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 2*x^17 + 6*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 32*x^12 - 48*x^11 + 80*x^10 - 96*x^9 + 128*x^8 - 192*x^7 + 288*x^6 - 384*x^5 + 384*x^4 - 256*x^3 + 256*x^2 - 1024*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} - 2 x^{17} + 6 x^{16} - 12 x^{15} + 18 x^{14} - 24 x^{13} + 32 x^{12} - 48 x^{11} + 80 x^{10} - 96 x^{9} + 128 x^{8} - 192 x^{7} + 288 x^{6} - 384 x^{5} + 384 x^{4} - 256 x^{3} + 256 x^{2} - 1024 x + 1024 \)

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:  \(11803130064401712301503252791296=2^{28}\cdot 13^{6}\cdot 457^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.78$
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:  $2, 13, 457$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{1}{8} a^{9} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{64} a^{14} + \frac{1}{64} a^{12} + \frac{1}{32} a^{10} - \frac{1}{32} a^{8} + \frac{3}{16} a^{7} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{15} + \frac{1}{128} a^{13} - \frac{1}{32} a^{12} + \frac{1}{64} a^{11} + \frac{1}{32} a^{10} - \frac{1}{64} a^{9} - \frac{1}{32} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} - \frac{7}{16} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{16} - \frac{1}{128} a^{14} - \frac{3}{64} a^{10} + \frac{3}{32} a^{9} + \frac{1}{32} a^{8} - \frac{3}{16} a^{7} + \frac{3}{16} a^{6} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{256} a^{17} - \frac{1}{256} a^{15} - \frac{3}{128} a^{11} + \frac{3}{64} a^{10} - \frac{7}{64} a^{9} - \frac{3}{32} a^{8} + \frac{7}{32} a^{7} - \frac{3}{16} a^{6} - \frac{5}{16} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{512} a^{18} - \frac{1}{512} a^{16} + \frac{5}{256} a^{12} + \frac{3}{128} a^{11} + \frac{5}{128} a^{10} - \frac{3}{64} a^{9} + \frac{7}{64} a^{8} + \frac{5}{32} a^{7} + \frac{1}{32} a^{6} + \frac{5}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{19} - \frac{1}{512} a^{17} - \frac{3}{256} a^{13} + \frac{3}{128} a^{12} + \frac{1}{128} a^{11} - \frac{3}{64} a^{10} + \frac{3}{64} a^{9} - \frac{3}{32} a^{8} + \frac{3}{32} a^{7} + \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{42}$, which has order $42$ (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:  \( \frac{3}{256} a^{19} - \frac{11}{256} a^{18} + \frac{3}{64} a^{17} - \frac{5}{256} a^{16} + \frac{1}{256} a^{15} - \frac{1}{32} a^{14} + \frac{15}{128} a^{13} - \frac{25}{128} a^{12} + \frac{31}{128} a^{11} - \frac{3}{16} a^{10} + \frac{15}{64} a^{9} - \frac{3}{16} a^{8} - \frac{7}{32} a^{7} + \frac{1}{4} a^{6} + \frac{5}{16} a^{5} - \frac{15}{8} a^{4} + \frac{7}{4} a^{3} + 3 a^{2} - \frac{25}{2} a + 12 \) (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:  \( 13727494.0011 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.10.858892093935616.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[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}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
457Data not computed