Properties

Label 20.0.68400816839...0976.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 798143^{2}$
Root discriminant $11.01$
Ramified primes $2, 798143$
Class number $1$
Class group Trivial
Galois Group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{18} \) \(\mathstrut +\mathstrut x^{16} \) \(\mathstrut +\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 4 x^{10} \) \(\mathstrut -\mathstrut x^{8} \) \(\mathstrut +\mathstrut 6 x^{6} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(684008168396450430976=2^{30}\cdot 798143^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.01$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 798143$
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 \(a\))

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

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:  \( \frac{1}{2} a^{19} - \frac{3}{2} a^{15} + \frac{5}{2} a^{13} + a^{11} - a^{9} - \frac{9}{2} a^{7} + 3 a^{5} + 4 a^{3} - 2 a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} + \frac{3}{2} a^{13} + a^{11} - 2 a^{9} - 2 a^{7} + 2 a^{5} + \frac{5}{2} a^{3} - a \),  \( \frac{1}{2} a^{18} + \frac{1}{4} a^{17} - a^{16} - \frac{3}{4} a^{15} + \frac{1}{2} a^{14} + a^{13} + \frac{3}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{9}{4} a^{8} - \frac{5}{4} a^{7} - \frac{3}{4} a^{6} + a^{5} + 3 a^{4} + \frac{3}{2} a^{3} - \frac{3}{2} a - \frac{1}{4} \),  \( \frac{1}{2} a^{19} + \frac{3}{4} a^{18} - \frac{1}{2} a^{17} - \frac{3}{2} a^{16} - \frac{1}{2} a^{15} + \frac{3}{2} a^{14} + \frac{3}{2} a^{13} + \frac{3}{4} a^{11} + \frac{5}{4} a^{10} - \frac{7}{4} a^{9} - \frac{5}{2} a^{8} - \frac{5}{2} a^{7} + \frac{7}{4} a^{5} + 2 a^{4} + \frac{9}{4} a^{3} - \frac{5}{4} a^{2} - \frac{5}{4} a + \frac{3}{4} \),  \( a^{19} - \frac{3}{2} a^{17} - \frac{3}{4} a^{16} + \frac{3}{2} a^{14} + \frac{9}{4} a^{13} - \frac{5}{4} a^{12} + \frac{5}{4} a^{11} - \frac{1}{2} a^{10} - 4 a^{9} - \frac{3}{4} a^{8} - \frac{13}{4} a^{7} + \frac{5}{2} a^{6} + \frac{19}{4} a^{5} + \frac{1}{2} a^{4} + \frac{9}{4} a^{3} - \frac{9}{4} a^{2} - \frac{3}{2} a + \frac{3}{4} \),  \( \frac{5}{4} a^{19} - 2 a^{17} - \frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{3}{4} a^{14} + 3 a^{13} - a^{12} + \frac{3}{4} a^{11} + \frac{1}{4} a^{10} - \frac{9}{2} a^{9} - \frac{1}{4} a^{8} - \frac{7}{2} a^{7} + \frac{5}{4} a^{6} + \frac{27}{4} a^{5} - a^{4} + \frac{7}{4} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + \frac{3}{2} \),  \( \frac{1}{4} a^{19} - \frac{3}{4} a^{18} - \frac{1}{4} a^{17} + \frac{3}{2} a^{16} - \frac{1}{4} a^{15} - a^{14} + a^{13} - a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - a^{9} + \frac{11}{4} a^{8} - \frac{3}{4} a^{7} + \frac{1}{2} a^{6} + \frac{7}{4} a^{5} - \frac{15}{4} a^{4} + \frac{1}{2} a^{3} + a^{2} - \frac{3}{2} a + \frac{1}{2} \),  \( \frac{1}{4} a^{19} + \frac{3}{4} a^{18} - \frac{1}{4} a^{17} - \frac{3}{2} a^{16} - \frac{1}{4} a^{15} + a^{14} + a^{13} + a^{12} + \frac{1}{4} a^{11} + \frac{1}{2} a^{10} - a^{9} - \frac{11}{4} a^{8} - \frac{3}{4} a^{7} - \frac{1}{2} a^{6} + \frac{7}{4} a^{5} + \frac{15}{4} a^{4} + \frac{1}{2} a^{3} - a^{2} - \frac{3}{2} a - \frac{1}{2} \),  \( \frac{1}{4} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} + \frac{1}{4} a^{16} + \frac{1}{2} a^{15} + a^{14} + \frac{1}{4} a^{13} - 2 a^{12} - \frac{3}{4} a^{10} - \frac{1}{2} a^{9} + \frac{5}{4} a^{8} + \frac{1}{4} a^{7} + \frac{15}{4} a^{6} + \frac{3}{4} a^{5} - \frac{9}{4} a^{4} - a^{3} - 3 a^{2} - \frac{1}{4} a + \frac{5}{4} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 171.360926564 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T1021:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.2.817298432.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
798143Data not computed