# Properties

 Label 20.0.96021059366...0576.1 Degree $20$ Signature $[0, 10]$ Discriminant $2^{20}\cdot 5501^{4}$ Root discriminant $11.20$ Ramified primes $2, 5501$ Class number $1$ Class group Trivial Galois Group 20T669

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 2, 0, 0, 0, 1, 0, 7, 0, -6, 0, 7, 0, 1, 0, 0, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 + x^14 + 7*x^12 - 6*x^10 + 7*x^8 + x^6 + 2*x^2 + 1)
gp: K = bnfinit(x^20 + 2*x^18 + x^14 + 7*x^12 - 6*x^10 + 7*x^8 + x^6 + 2*x^2 + 1, 1)

## Normalizeddefining polynomial

$$x^{20}$$ $$\mathstrut +\mathstrut 2 x^{18}$$ $$\mathstrut +\mathstrut x^{14}$$ $$\mathstrut +\mathstrut 7 x^{12}$$ $$\mathstrut -\mathstrut 6 x^{10}$$ $$\mathstrut +\mathstrut 7 x^{8}$$ $$\mathstrut +\mathstrut x^{6}$$ $$\mathstrut +\mathstrut 2 x^{2}$$ $$\mathstrut +\mathstrut 1$$

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: $$960210593667613720576=2^{20}\cdot 5501^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.20$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 5501$ 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 $$a$$)

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

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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a^{19} + 2 a^{17} - \frac{3}{2} a^{15} - \frac{5}{2} a^{13} + 7 a^{11} - \frac{9}{2} a^{9} - \frac{5}{2} a^{7} + 5 a^{5} - \frac{5}{2} a^{3} - \frac{3}{2} a$$,  $$\frac{5}{8} a^{19} + \frac{13}{8} a^{17} + \frac{3}{8} a^{15} - \frac{3}{4} a^{13} + \frac{25}{8} a^{11} - \frac{15}{8} a^{9} + \frac{1}{4} a^{7} + \frac{19}{8} a^{5} - \frac{3}{8} a^{3} + \frac{5}{8} a$$,  $$a$$,  $$\frac{1}{8} a^{19} + \frac{9}{8} a^{17} + \frac{19}{8} a^{15} + \frac{3}{4} a^{13} - \frac{3}{8} a^{11} + \frac{33}{8} a^{9} - \frac{1}{4} a^{7} - \frac{9}{8} a^{5} + \frac{29}{8} a^{3} + \frac{5}{8} a$$,  $$\frac{1}{16} a^{19} - \frac{7}{8} a^{18} + \frac{3}{16} a^{17} - \frac{5}{4} a^{16} - \frac{9}{16} a^{15} + \frac{13}{8} a^{14} - \frac{3}{2} a^{13} + \frac{3}{8} a^{12} + \frac{11}{16} a^{11} - \frac{13}{2} a^{10} + \frac{17}{16} a^{9} + \frac{57}{8} a^{8} - 4 a^{7} - \frac{41}{8} a^{6} + \frac{45}{16} a^{5} + \frac{1}{4} a^{4} - \frac{7}{16} a^{3} + \frac{3}{8} a^{2} - \frac{21}{16} a$$,  $$\frac{1}{8} a^{19} + \frac{23}{16} a^{18} + \frac{7}{4} a^{17} + \frac{67}{16} a^{16} + \frac{31}{8} a^{15} + \frac{33}{16} a^{14} + \frac{9}{8} a^{13} - \frac{11}{8} a^{12} - \frac{1}{2} a^{11} + \frac{119}{16} a^{10} + \frac{63}{8} a^{9} + \frac{3}{16} a^{8} - \frac{11}{8} a^{7} + \frac{5}{8} a^{6} + \frac{9}{4} a^{5} + \frac{61}{16} a^{4} + \frac{25}{8} a^{3} + \frac{31}{16} a^{2} + \frac{9}{4} a + \frac{11}{16}$$,  $$\frac{1}{8} a^{19} + \frac{1}{2} a^{18} + \frac{7}{8} a^{17} + \frac{7}{8} a^{16} + \frac{21}{8} a^{15} - \frac{1}{2} a^{14} + \frac{11}{4} a^{13} - \frac{3}{8} a^{12} - \frac{1}{8} a^{11} + \frac{21}{8} a^{10} + \frac{11}{8} a^{9} - \frac{15}{4} a^{8} + \frac{23}{4} a^{7} + \frac{25}{8} a^{6} - \frac{23}{8} a^{5} - \frac{7}{8} a^{4} + \frac{27}{8} a^{3} + \frac{1}{2} a^{2} + \frac{21}{8} a - \frac{1}{8}$$,  $$\frac{1}{16} a^{19} + \frac{1}{16} a^{18} + \frac{3}{16} a^{17} + \frac{13}{16} a^{16} + \frac{3}{16} a^{15} + \frac{31}{16} a^{14} + \frac{1}{4} a^{13} + \frac{7}{8} a^{12} + \frac{11}{16} a^{11} - \frac{7}{16} a^{10} + \frac{5}{16} a^{9} + \frac{45}{16} a^{8} + \frac{3}{4} a^{7} - \frac{1}{8} a^{6} + \frac{13}{16} a^{5} + \frac{3}{16} a^{4} + \frac{13}{16} a^{3} + \frac{17}{16} a^{2} + \frac{15}{16} a + \frac{21}{16}$$,  $$\frac{1}{16} a^{19} + \frac{3}{2} a^{18} + \frac{3}{16} a^{17} + \frac{29}{8} a^{16} - \frac{9}{16} a^{15} - \frac{1}{4} a^{14} - \frac{3}{2} a^{13} - \frac{23}{8} a^{12} + \frac{11}{16} a^{11} + \frac{71}{8} a^{10} + \frac{17}{16} a^{9} - \frac{7}{2} a^{8} - 4 a^{7} - \frac{11}{8} a^{6} + \frac{45}{16} a^{5} + \frac{43}{8} a^{4} - \frac{7}{16} a^{3} - \frac{3}{4} a^{2} - \frac{21}{16} a - \frac{13}{8}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$103.153871476$$ 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 61440 The 126 conjugacy class representatives for t20n669 are not computed Character table for t20n669 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 siblings: data not computed Degree 40 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4} 2.12.12.11x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
5501Data not computed