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Properties

Label 20.4.25023434548...6256.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{32}\cdot 17^{12}$
Root discriminant $16.59$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_2\times A_5$ (as 20T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -32, 0, -84, 0, -134, 0, -146, 0, -102, 0, -43, 0, -2, 0, 11, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 + 11*x^16 - 2*x^14 - 43*x^12 - 102*x^10 - 146*x^8 - 134*x^6 - 84*x^4 - 32*x^2 + 1)
 
gp: K = bnfinit(x^20 + 6*x^18 + 11*x^16 - 2*x^14 - 43*x^12 - 102*x^10 - 146*x^8 - 134*x^6 - 84*x^4 - 32*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 6 x^{18} + 11 x^{16} - 2 x^{14} - 43 x^{12} - 102 x^{10} - 146 x^{8} - 134 x^{6} - 84 x^{4} - 32 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2502343454824177132896256=2^{32}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.59$
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:  $2, 17$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28505.6799029 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

10.2.98867482624.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.31$x^{12} + 14 x^{11} + 4 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} - 8 x^{5} + 4 x^{4} + 8 x^{3} + 16 x^{2} + 16 x - 8$$4$$3$$18$$A_4\times C_2$$[2, 2, 2]^{3}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$