Properties

Label 20.0.16182995430...8304.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 277^{4}$
Root discriminant $11.49$
Ramified primes $2, 277$
Class number $1$
Class group Trivial
Galois Group 20T279

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\(x^{20} \) \(\mathstrut +\mathstrut 4 x^{16} \) \(\mathstrut -\mathstrut 4 x^{12} \) \(\mathstrut +\mathstrut 4 x^{8} \) \(\mathstrut -\mathstrut 8 x^{4} \) \(\mathstrut +\mathstrut 4 \)

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:  \(1618299543010938978304=2^{38}\cdot 277^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.49$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 277$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{20} a^{16} + \frac{1}{10} a^{12} - \frac{2}{5} a^{8} - \frac{1}{2} a^{6} - \frac{2}{5}$, $\frac{1}{20} a^{17} + \frac{1}{10} a^{13} - \frac{2}{5} a^{9} - \frac{1}{2} a^{7} - \frac{2}{5} a$, $\frac{1}{20} a^{18} + \frac{1}{10} a^{14} + \frac{1}{10} a^{10} - \frac{1}{2} a^{8} - \frac{2}{5} a^{2}$, $\frac{1}{20} a^{19} + \frac{1}{10} a^{15} + \frac{1}{10} a^{11} - \frac{1}{2} a^{9} - \frac{2}{5} a^{3}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -\frac{13}{10} a^{18} - \frac{61}{10} a^{14} + \frac{9}{10} a^{10} - 5 a^{6} + \frac{32}{5} a^{2} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{3}{5} a^{19} - \frac{2}{5} a^{17} + \frac{59}{20} a^{15} - \frac{9}{5} a^{13} + \frac{1}{5} a^{11} + \frac{7}{10} a^{9} + 2 a^{7} - \frac{3}{2} a^{5} - \frac{14}{5} a^{3} + \frac{11}{5} a \),  \( \frac{13}{10} a^{18} + \frac{61}{10} a^{14} - \frac{9}{10} a^{10} + 5 a^{6} - \frac{37}{5} a^{2} \),  \( \frac{1}{20} a^{19} - \frac{11}{20} a^{17} + \frac{7}{20} a^{15} - \frac{13}{5} a^{13} + \frac{3}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{2} a^{7} - \frac{3}{2} a^{5} - \frac{2}{5} a^{3} + \frac{17}{5} a \),  \( \frac{1}{10} a^{18} + \frac{9}{10} a^{17} + \frac{1}{10} a^{16} + \frac{9}{20} a^{14} + \frac{43}{10} a^{13} + \frac{9}{20} a^{12} - \frac{1}{20} a^{10} - \frac{1}{5} a^{9} - \frac{3}{10} a^{8} + a^{6} + \frac{7}{2} a^{5} - \frac{3}{10} a^{2} - \frac{26}{5} a - \frac{3}{10} \),  \( \frac{5}{4} a^{19} + \frac{13}{20} a^{18} + \frac{9}{20} a^{17} + \frac{3}{5} a^{16} + 6 a^{15} + \frac{61}{20} a^{14} + \frac{43}{20} a^{13} + \frac{59}{20} a^{12} - \frac{1}{4} a^{11} - \frac{9}{20} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{9}{2} a^{7} + \frac{5}{2} a^{6} + 2 a^{5} + 2 a^{4} - \frac{13}{2} a^{3} - \frac{37}{10} a^{2} - \frac{21}{10} a - \frac{33}{10} \),  \( \frac{5}{4} a^{19} - \frac{13}{20} a^{18} + \frac{9}{20} a^{17} - \frac{3}{5} a^{16} + 6 a^{15} - \frac{61}{20} a^{14} + \frac{43}{20} a^{13} - \frac{59}{20} a^{12} - \frac{1}{4} a^{11} + \frac{9}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{9}{2} a^{7} - \frac{5}{2} a^{6} + 2 a^{5} - 2 a^{4} - \frac{13}{2} a^{3} + \frac{37}{10} a^{2} - \frac{21}{10} a + \frac{33}{10} \),  \( \frac{5}{4} a^{19} - \frac{1}{10} a^{18} - \frac{9}{20} a^{17} + \frac{9}{20} a^{16} + 6 a^{15} - \frac{9}{20} a^{14} - \frac{43}{20} a^{13} + \frac{43}{20} a^{12} - \frac{1}{4} a^{11} + \frac{1}{20} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{9}{2} a^{7} - a^{6} - 2 a^{5} + 2 a^{4} - \frac{13}{2} a^{3} + \frac{3}{10} a^{2} + \frac{21}{10} a - \frac{21}{10} \),  \( \frac{13}{20} a^{19} + \frac{13}{20} a^{18} - \frac{1}{20} a^{17} - \frac{3}{5} a^{16} + \frac{61}{20} a^{15} + \frac{61}{20} a^{14} - \frac{7}{20} a^{13} - \frac{59}{20} a^{12} - \frac{9}{20} a^{11} - \frac{9}{20} a^{10} - \frac{3}{5} a^{9} - \frac{1}{5} a^{8} + \frac{5}{2} a^{7} + \frac{5}{2} a^{6} - \frac{1}{2} a^{5} - 2 a^{4} - \frac{37}{10} a^{3} - \frac{37}{10} a^{2} - \frac{1}{10} a + \frac{33}{10} \),  \( \frac{1}{20} a^{18} + \frac{7}{20} a^{17} + a^{16} + \frac{7}{20} a^{14} + \frac{17}{10} a^{13} + \frac{19}{4} a^{12} + \frac{3}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} + \frac{1}{2} a^{6} + 2 a^{5} + \frac{7}{2} a^{4} + \frac{1}{10} a^{2} - \frac{9}{5} a - 5 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 476.032179845 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T279:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.4432.1, 10.0.1257127936.1, 10.2.5028511744.1, 10.0.5028511744.1

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 20 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
277Data not computed