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Properties

Label 20.0.54798626427...9504.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 3^{2}\cdot 881^{8}$
Root discriminant $38.63$
Ramified primes $2, 3, 881$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 13, 0, 8, 0, -85, 0, 187, 0, -232, 0, 187, 0, -85, 0, 8, 0, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 13*x^18 + 8*x^16 - 85*x^14 + 187*x^12 - 232*x^10 + 187*x^8 - 85*x^6 + 8*x^4 + 13*x^2 + 1)
 
gp: K = bnfinit(x^20 + 13*x^18 + 8*x^16 - 85*x^14 + 187*x^12 - 232*x^10 + 187*x^8 - 85*x^6 + 8*x^4 + 13*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 13 x^{18} + 8 x^{16} - 85 x^{14} + 187 x^{12} - 232 x^{10} + 187 x^{8} - 85 x^{6} + 8 x^{4} + 13 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54798626427212061103203500949504=2^{24}\cdot 3^{2}\cdot 881^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.63$
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:  $2, 3, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{1}{4} a^{3} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{768} a^{16} - \frac{5}{768} a^{14} + \frac{97}{768} a^{12} + \frac{47}{384} a^{10} - \frac{11}{128} a^{8} - \frac{1}{2} a^{7} - \frac{145}{384} a^{6} - \frac{287}{768} a^{4} - \frac{5}{768} a^{2} - \frac{1}{2} a + \frac{1}{768}$, $\frac{1}{768} a^{17} - \frac{5}{768} a^{15} + \frac{97}{768} a^{13} + \frac{47}{384} a^{11} - \frac{11}{128} a^{9} + \frac{47}{384} a^{7} - \frac{1}{2} a^{6} - \frac{287}{768} a^{5} - \frac{5}{768} a^{3} - \frac{383}{768} a - \frac{1}{2}$, $\frac{1}{6144} a^{18} + \frac{3}{256} a^{14} - \frac{447}{2048} a^{12} - \frac{187}{1536} a^{10} + \frac{37}{1536} a^{8} - \frac{1}{2} a^{7} + \frac{61}{2048} a^{6} + \frac{5}{64} a^{4} + \frac{127}{256} a^{2} - \frac{1}{2} a - \frac{3067}{6144}$, $\frac{1}{6144} a^{19} + \frac{3}{256} a^{15} - \frac{447}{2048} a^{13} - \frac{187}{1536} a^{11} + \frac{37}{1536} a^{9} - \frac{963}{2048} a^{7} - \frac{1}{2} a^{6} + \frac{5}{64} a^{5} + \frac{127}{256} a^{3} + \frac{5}{6144} a - \frac{1}{2}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( \frac{407}{768} a^{19} + \frac{2633}{384} a^{17} + \frac{1487}{384} a^{15} - \frac{34217}{768} a^{13} + \frac{9853}{96} a^{11} - \frac{12827}{96} a^{9} + \frac{88333}{768} a^{7} - \frac{22663}{384} a^{5} + \frac{4799}{384} a^{3} + \frac{3845}{768} a \) (order $4$)
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:  \( 23467601.0763 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.3104644.1, 10.6.7402609433653248.1, 10.4.28916443100208.1, 10.0.2467536477884416.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
881Data not computed