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Properties

Label 20.12.4617939718...4576.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{18}\cdot 89^{2}$
Root discriminant $27.12$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 22, 170, 814, 2423, 3806, 1115, -5984, -8692, -1122, 7041, 5456, -644, -2442, -679, 396, 213, -22, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 24*x^18 - 22*x^17 + 213*x^16 + 396*x^15 - 679*x^14 - 2442*x^13 - 644*x^12 + 5456*x^11 + 7041*x^10 - 1122*x^9 - 8692*x^8 - 5984*x^7 + 1115*x^6 + 3806*x^5 + 2423*x^4 + 814*x^3 + 170*x^2 + 22*x + 1)
 
gp: K = bnfinit(x^20 - 24*x^18 - 22*x^17 + 213*x^16 + 396*x^15 - 679*x^14 - 2442*x^13 - 644*x^12 + 5456*x^11 + 7041*x^10 - 1122*x^9 - 8692*x^8 - 5984*x^7 + 1115*x^6 + 3806*x^5 + 2423*x^4 + 814*x^3 + 170*x^2 + 22*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 24 x^{18} - 22 x^{17} + 213 x^{16} + 396 x^{15} - 679 x^{14} - 2442 x^{13} - 644 x^{12} + 5456 x^{11} + 7041 x^{10} - 1122 x^{9} - 8692 x^{8} - 5984 x^{7} + 1115 x^{6} + 3806 x^{5} + 2423 x^{4} + 814 x^{3} + 170 x^{2} + 22 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46179397182603358960092184576=2^{20}\cdot 11^{18}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.12$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{14} + \frac{1}{11} a^{13} - \frac{2}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{2}{11} a^{8} - \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{3}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} + \frac{2}{11} a^{13} - \frac{5}{11} a^{12} + \frac{4}{11} a^{10} + \frac{1}{11} a^{9} + \frac{3}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{17} - \frac{3}{11} a^{14} - \frac{1}{11} a^{13} + \frac{3}{11} a^{12} + \frac{5}{11} a^{11} - \frac{3}{11} a^{10} + \frac{2}{11} a^{9} - \frac{5}{11} a^{8} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} - \frac{3}{11} a^{5} + \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{18} - \frac{2}{11} a^{14} - \frac{5}{11} a^{13} - \frac{1}{11} a^{12} - \frac{5}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} - \frac{3}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{34029126729763} a^{19} - \frac{706478636332}{34029126729763} a^{18} + \frac{138106245273}{3093556975433} a^{17} + \frac{316518967966}{34029126729763} a^{16} - \frac{3945349472}{110844061009} a^{15} + \frac{5008230971213}{34029126729763} a^{14} - \frac{6240856718859}{34029126729763} a^{13} + \frac{8111656572754}{34029126729763} a^{12} - \frac{2184182067517}{34029126729763} a^{11} - \frac{3786530185312}{34029126729763} a^{10} + \frac{3421020492883}{34029126729763} a^{9} - \frac{694680895922}{34029126729763} a^{8} - \frac{5364274894295}{34029126729763} a^{7} + \frac{10148202901395}{34029126729763} a^{6} + \frac{6664600577169}{34029126729763} a^{5} + \frac{10158997036308}{34029126729763} a^{4} + \frac{16141329076806}{34029126729763} a^{3} - \frac{9710134811879}{34029126729763} a^{2} - \frac{3153943010313}{34029126729763} a + \frac{15686073374523}{34029126729763}$

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:  $15$
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:  \( 13201238.9695 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$