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Properties

Label 20.0.53793575578...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{12}\cdot 13^{15}$
Root discriminant $43.30$
Ramified primes $3, 5, 13$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 0, 0, 0, 0, -12393, 0, 0, 0, 0, 1107, 0, 0, 0, 0, -51, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 51*x^15 + 1107*x^10 - 12393*x^5 + 59049)
 
gp: K = bnfinit(x^20 - 51*x^15 + 1107*x^10 - 12393*x^5 + 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 51 x^{15} + 1107 x^{10} - 12393 x^{5} + 59049 \)

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:  \(537935755789407686830516845703125=3^{16}\cdot 5^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.30$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{9} a^{8} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{4}$, $\frac{1}{27} a^{10} + \frac{1}{9} a^{5}$, $\frac{1}{27} a^{11} + \frac{1}{9} a^{6}$, $\frac{1}{81} a^{12} + \frac{1}{27} a^{7} - \frac{1}{3} a^{2}$, $\frac{1}{81} a^{13} + \frac{1}{27} a^{8} - \frac{1}{3} a^{3}$, $\frac{1}{243} a^{14} + \frac{1}{81} a^{9} + \frac{2}{9} a^{4}$, $\frac{1}{729} a^{15} + \frac{1}{243} a^{10} + \frac{2}{27} a^{5}$, $\frac{1}{3645} a^{16} + \frac{1}{3645} a^{15} - \frac{2}{1215} a^{14} - \frac{1}{405} a^{12} + \frac{1}{1215} a^{11} + \frac{2}{243} a^{10} - \frac{2}{405} a^{9} + \frac{2}{45} a^{8} + \frac{8}{135} a^{7} + \frac{4}{27} a^{6} - \frac{22}{135} a^{5} - \frac{4}{45} a^{4} + \frac{1}{3} a^{3} + \frac{7}{15} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{10935} a^{17} + \frac{1}{3645} a^{15} - \frac{1}{1215} a^{14} - \frac{2}{405} a^{13} + \frac{19}{3645} a^{12} + \frac{2}{135} a^{11} - \frac{17}{1215} a^{10} - \frac{2}{81} a^{9} + \frac{4}{135} a^{8} - \frac{28}{405} a^{7} + \frac{2}{45} a^{6} + \frac{1}{27} a^{5} + \frac{13}{45} a^{4} + \frac{4}{15} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{32805} a^{18} + \frac{2}{3645} a^{15} + \frac{2}{1215} a^{14} + \frac{64}{10935} a^{13} - \frac{1}{405} a^{12} + \frac{1}{135} a^{11} + \frac{4}{243} a^{10} - \frac{7}{405} a^{9} + \frac{14}{1215} a^{8} - \frac{19}{135} a^{7} - \frac{1}{9} a^{6} - \frac{8}{135} a^{5} - \frac{8}{45} a^{4} + \frac{4}{45} a^{3} - \frac{2}{15} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{98415} a^{19} - \frac{2}{3645} a^{15} + \frac{2}{6561} a^{14} - \frac{2}{405} a^{13} - \frac{1}{135} a^{11} - \frac{2}{1215} a^{10} + \frac{131}{3645} a^{9} - \frac{2}{135} a^{8} + \frac{2}{15} a^{7} - \frac{7}{45} a^{6} + \frac{1}{27} a^{5} + \frac{52}{135} a^{4} - \frac{4}{15} a^{3} + \frac{2}{5} a - \frac{1}{5}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( -1 \) (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:  \( 40743102.9736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.5.22244625.1, 10.10.6432703438078125.1

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 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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed
13Data not computed