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Properties

Label 20.4.43601184876...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{40}\cdot 5^{10}\cdot 67^{8}$
Root discriminant $48.08$
Ramified primes $2, 5, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 3024, 0, 11585, 0, 16884, 0, 7790, 0, -3648, 0, -4669, 0, -1392, 0, -82, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 - 82*x^16 - 1392*x^14 - 4669*x^12 - 3648*x^10 + 7790*x^8 + 16884*x^6 + 11585*x^4 + 3024*x^2 + 256)
 
gp: K = bnfinit(x^20 + 12*x^18 - 82*x^16 - 1392*x^14 - 4669*x^12 - 3648*x^10 + 7790*x^8 + 16884*x^6 + 11585*x^4 + 3024*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} - 82 x^{16} - 1392 x^{14} - 4669 x^{12} - 3648 x^{10} + 7790 x^{8} + 16884 x^{6} + 11585 x^{4} + 3024 x^{2} + 256 \)

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:  \(4360118487671115706531840000000000=2^{40}\cdot 5^{10}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.08$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{40} a^{16} - \frac{1}{20} a^{14} - \frac{3}{40} a^{12} + \frac{3}{40} a^{10} + \frac{11}{40} a^{8} + \frac{1}{40} a^{6} - \frac{1}{2} a^{5} - \frac{9}{20} a^{4} - \frac{1}{2} a^{3} + \frac{17}{40} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{40} a^{17} - \frac{1}{20} a^{15} - \frac{3}{40} a^{13} + \frac{3}{40} a^{11} + \frac{1}{40} a^{9} - \frac{19}{40} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{3}{40} a^{3} - \frac{1}{2} a^{2} + \frac{7}{20} a$, $\frac{1}{6825162615520} a^{18} - \frac{4416673663}{426572663470} a^{16} + \frac{82450421647}{3412581307760} a^{14} - \frac{1}{8} a^{13} - \frac{18626261231}{1706290653880} a^{12} - \frac{1}{8} a^{11} + \frac{101532419143}{6825162615520} a^{10} + \frac{122574339567}{341258130776} a^{8} - \frac{3}{8} a^{7} + \frac{285660294973}{3412581307760} a^{6} - \frac{141830819761}{341258130776} a^{4} - \frac{1}{8} a^{3} - \frac{3144078426763}{6825162615520} a^{2} - \frac{1}{8} a + \frac{83932725031}{426572663470}$, $\frac{1}{27300650462080} a^{19} + \frac{4998114339}{1365032523104} a^{17} + \frac{764966683199}{13650325231040} a^{15} - \frac{9162378767}{426572663470} a^{13} + \frac{2319710269187}{27300650462080} a^{11} + \frac{28619537589}{853145326940} a^{9} + \frac{6342992116247}{13650325231040} a^{7} - \frac{1}{2} a^{6} - \frac{623839566111}{6825162615520} a^{5} + \frac{3169196992593}{27300650462080} a^{3} + \frac{63294995689}{853145326940} a - \frac{1}{2}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 719958084.406 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.5745920.1, 10.10.4126949580800000.1, 10.2.528249546342400.1, 10.2.66031193292800000.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$