# Properties

 Label 20.0.36235542197...7561.1 Degree $20$ Signature $[0, 10]$ Discriminant $47^{10}\cdot 83^{2}$ Root discriminant $10.66$ Ramified primes $47, 83$ Class number $1$ Class group Trivial Galois group $C_2\times C_2^4:D_5$ (as 20T81)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 4, 1, -8, 8, 17, -13, 3, 3, -3, 8, -1, 8, -7, 5, -4, 3, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1)

gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1, 1)

## Normalizeddefining polynomial

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

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: $$362355421972633207561=47^{10}\cdot 83^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.66$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $47, 83$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{2194608515} a^{19} + \frac{74558294}{2194608515} a^{18} - \frac{1017294346}{2194608515} a^{17} + \frac{6939284}{39901973} a^{16} + \frac{856912424}{2194608515} a^{15} - \frac{994013257}{2194608515} a^{14} - \frac{586393183}{2194608515} a^{13} - \frac{382352308}{2194608515} a^{12} + \frac{9694604}{438921703} a^{11} - \frac{654463201}{2194608515} a^{10} + \frac{509560233}{2194608515} a^{9} + \frac{127710497}{2194608515} a^{8} - \frac{19417716}{438921703} a^{7} + \frac{941830149}{2194608515} a^{6} - \frac{675824012}{2194608515} a^{5} - \frac{811533444}{2194608515} a^{4} - \frac{1050222631}{2194608515} a^{3} + \frac{195415228}{438921703} a^{2} + \frac{339804044}{2194608515} a - \frac{29307092}{438921703}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has 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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$59.602445011$$ 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 solvable group of order 320 The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$ Character table for $C_2\times C_2^4:D_5$

## 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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 47.4.2.1x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 47.4.2.1x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2} 83.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 83.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 83.2.1.2x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$