# Properties

 Label 20.0.35754806217...3081.1 Degree $20$ Signature $[0, 10]$ Discriminant $3^{10}\cdot 7^{10}\cdot 11^{8}$ Root discriminant $11.96$ Ramified primes $3, 7, 11$ Class number $1$ Class group Trivial Galois group $C_{10}\times D_5$ (as 20T24)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 6, -10, 5, 1, -6, 20, -26, 10, 6, -1, -13, 18, -9, -2, 3, 2, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1)

gp: K = bnfinit(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1, 1)

## Normalizeddefining polynomial

$$x^{20} - 3 x^{19} + 2 x^{18} + 3 x^{17} - 2 x^{16} - 9 x^{15} + 18 x^{14} - 13 x^{13} - x^{12} + 6 x^{11} + 10 x^{10} - 26 x^{9} + 20 x^{8} - 6 x^{7} + x^{6} + 5 x^{5} - 10 x^{4} + 6 x^{3} - 2 x + 1$$

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: $$3575480621700351753081=3^{10}\cdot 7^{10}\cdot 11^{8}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.96$ 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: $3, 7, 11$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $10$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2663593} a^{19} - \frac{605577}{2663593} a^{18} + \frac{865553}{2663593} a^{17} - \frac{1243914}{2663593} a^{16} - \frac{768917}{2663593} a^{15} + \frac{133054}{2663593} a^{14} - \frac{354728}{2663593} a^{13} + \frac{605595}{2663593} a^{12} - \frac{1111512}{2663593} a^{11} - \frac{501171}{2663593} a^{10} + \frac{1013558}{2663593} a^{9} + \frac{680637}{2663593} a^{8} - \frac{1035426}{2663593} a^{7} + \frac{1290760}{2663593} a^{6} - \frac{685238}{2663593} a^{5} + \frac{1163147}{2663593} a^{4} - \frac{394096}{2663593} a^{3} - \frac{978097}{2663593} a^{2} - \frac{1053511}{2663593} a + \frac{402138}{2663593}$

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: $$-\frac{5069577}{2663593} a^{19} + \frac{15210989}{2663593} a^{18} - \frac{13772404}{2663593} a^{17} - \frac{5212947}{2663593} a^{16} + \frac{3481178}{2663593} a^{15} + \frac{38083464}{2663593} a^{14} - \frac{89138870}{2663593} a^{13} + \frac{95969693}{2663593} a^{12} - \frac{52519168}{2663593} a^{11} + \frac{16837722}{2663593} a^{10} - \frac{63630828}{2663593} a^{9} + \frac{132396579}{2663593} a^{8} - \frac{146687436}{2663593} a^{7} + \frac{116295184}{2663593} a^{6} - \frac{79084843}{2663593} a^{5} + \frac{20213325}{2663593} a^{4} + \frac{18815882}{2663593} a^{3} - \frac{27982575}{2663593} a^{2} + \frac{18220722}{2663593} a - \frac{3418100}{2663593}$$ (order $6$) 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: $$635.552521857$$ 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 100 The 40 conjugacy class representatives for $C_{10}\times D_5$ Character table for $C_{10}\times D_5$ 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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5} 7.10.5.2x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10} 11.10.8.1x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$