# 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

# Learn more about

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)

## Normalizeddefining 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

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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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