# Properties

 Label 20.0.29411962588...3449.1 Degree $20$ Signature $[0, 10]$ Discriminant $11^{18}\cdot 23^{2}$ Root discriminant $11.84$ Ramified primes $11, 23$ Class number $1$ Class group Trivial Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -72, 177, -363, 637, -973, 1307, -1556, 1649, -1556, 1307, -973, 637, -363, 177, -72, 24, -6, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1)

gp: K = bnfinit(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1, 1)

## Normalizeddefining polynomial

$$x^{20} - 6 x^{19} + 24 x^{18} - 72 x^{17} + 177 x^{16} - 363 x^{15} + 637 x^{14} - 973 x^{13} + 1307 x^{12} - 1556 x^{11} + 1649 x^{10} - 1556 x^{9} + 1307 x^{8} - 973 x^{7} + 637 x^{6} - 363 x^{5} + 177 x^{4} - 72 x^{3} + 24 x^{2} - 6 x + 1$$

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: $$2941196258837390453449=11^{18}\cdot 23^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.84$ 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: $11, 23$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109} a^{19} - \frac{7}{109} a^{18} + \frac{31}{109} a^{17} + \frac{6}{109} a^{16} - \frac{47}{109} a^{15} + \frac{11}{109} a^{14} - \frac{28}{109} a^{13} + \frac{36}{109} a^{12} - \frac{37}{109} a^{11} + \frac{7}{109} a^{10} + \frac{7}{109} a^{9} - \frac{37}{109} a^{8} + \frac{36}{109} a^{7} - \frac{28}{109} a^{6} + \frac{11}{109} a^{5} - \frac{47}{109} a^{4} + \frac{6}{109} a^{3} + \frac{31}{109} a^{2} - \frac{7}{109} a + \frac{1}{109}$

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: $$\frac{20}{109} a^{19} - \frac{140}{109} a^{18} + \frac{511}{109} a^{17} - \frac{1515}{109} a^{16} + \frac{3638}{109} a^{15} - \frac{7410}{109} a^{14} + \frac{12956}{109} a^{13} - \frac{20208}{109} a^{12} + \frac{28254}{109} a^{11} - \frac{34849}{109} a^{10} + \frac{38399}{109} a^{9} - \frac{37364}{109} a^{8} + \frac{31894}{109} a^{7} - \frac{23995}{109} a^{6} + \frac{15480}{109} a^{5} - \frac{8461}{109} a^{4} + \frac{4153}{109} a^{3} - \frac{1669}{109} a^{2} + \frac{623}{109} a - \frac{198}{109}$$ (order $22$) 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: $$2189.13991382$$ 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 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$ Character table for $C_2^2\times C_2^4:C_5$ 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2} 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$