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Properties

Label 20.4.11939665233...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}$
Root discriminant $25.34$
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![1024, -1536, 256, 256, -256, 128, 384, -320, 128, 0, -80, 0, 32, -40, 24, 4, -4, 2, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + x^18 + 2*x^17 - 4*x^16 + 4*x^15 + 24*x^14 - 40*x^13 + 32*x^12 - 80*x^10 + 128*x^8 - 320*x^7 + 384*x^6 + 128*x^5 - 256*x^4 + 256*x^3 + 256*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(x^20 - 3*x^19 + x^18 + 2*x^17 - 4*x^16 + 4*x^15 + 24*x^14 - 40*x^13 + 32*x^12 - 80*x^10 + 128*x^8 - 320*x^7 + 384*x^6 + 128*x^5 - 256*x^4 + 256*x^3 + 256*x^2 - 1536*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + x^{18} + 2 x^{17} - 4 x^{16} + 4 x^{15} + 24 x^{14} - 40 x^{13} + 32 x^{12} - 80 x^{10} + 128 x^{8} - 320 x^{7} + 384 x^{6} + 128 x^{5} - 256 x^{4} + 256 x^{3} + 256 x^{2} - 1536 x + 1024 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11939665233708135040000000000=2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.34$
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 \(a\))

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

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:  $11$
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:  \( 2674605.76576 \) (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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{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}$
$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}$