# Properties

 Label 20.0.33545186845...0752.1 Degree $20$ Signature $[0, 10]$ Discriminant $2^{16}\cdot 13^{15}$ Root discriminant $11.92$ Ramified primes $2, 13$ Class number $1$ Class group Trivial Galois group $F_5$ (as 20T5)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 10, -15, 0, 40, -64, 46, 8, -32, 8, 46, -64, 40, 0, -15, 10, 0, -2, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^17 - 15*x^16 + 40*x^14 - 64*x^13 + 46*x^12 + 8*x^11 - 32*x^10 + 8*x^9 + 46*x^8 - 64*x^7 + 40*x^6 - 15*x^4 + 10*x^3 - 2*x + 1)

gp: K = bnfinit(x^20 - 2*x^19 + 10*x^17 - 15*x^16 + 40*x^14 - 64*x^13 + 46*x^12 + 8*x^11 - 32*x^10 + 8*x^9 + 46*x^8 - 64*x^7 + 40*x^6 - 15*x^4 + 10*x^3 - 2*x + 1, 1)

## Normalizeddefining polynomial

$$x^{20} - 2 x^{19} + 10 x^{17} - 15 x^{16} + 40 x^{14} - 64 x^{13} + 46 x^{12} + 8 x^{11} - 32 x^{10} + 8 x^{9} + 46 x^{8} - 64 x^{7} + 40 x^{6} - 15 x^{4} + 10 x^{3} - 2 x + 1$$

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: $$3354518684571451850752=2^{16}\cdot 13^{15}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.92$ 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: $2, 13$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Gal(K/\Q)|$: $20$ This field is Galois over $\Q$. This is not a CM field.

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

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

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: $$238.906276128$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 20 The 5 conjugacy class representatives for $F_5$ Character table for $F_5$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 5 sibling: 5.1.35152.1 Degree 10 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4} 13.4.3.2x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4} 13.4.3.2x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$