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

Related objects

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)

Normalizeddefining 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

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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 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