# Properties

 Label 20.0.26762285357...3376.1 Degree $20$ Signature $[0, 10]$ Discriminant $2^{20}\cdot 761^{5}$ Root discriminant $10.50$ Ramified primes $2, 761$ Class number $1$ Class group Trivial Galois group $D_5\wr C_2$ (as 20T48)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1)

gp: K = bnfinit(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 27, -54, 88, -144, 188, -140, 44, -34, 92, -92, 25, 4, 17, -18, 6, 0, 0, -2, 1]);

## Normalizeddefining polynomial

$$x^{20} - 2 x^{19} + 6 x^{16} - 18 x^{15} + 17 x^{14} + 4 x^{13} + 25 x^{12} - 92 x^{11} + 92 x^{10} - 34 x^{9} + 44 x^{8} - 140 x^{7} + 188 x^{6} - 144 x^{5} + 88 x^{4} - 54 x^{3} + 27 x^{2} - 8 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $20$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 10]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$267622853577068773376=2^{20}\cdot 761^{5}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $10.50$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 761$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $10$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{4379139123} a^{19} - \frac{119017966}{4379139123} a^{18} + \frac{1685921074}{4379139123} a^{17} - \frac{2114361461}{4379139123} a^{16} + \frac{33058990}{1459713041} a^{15} + \frac{365282}{257596419} a^{14} + \frac{114632567}{257596419} a^{13} - \frac{973911976}{4379139123} a^{12} - \frac{1115738719}{4379139123} a^{11} + \frac{537527623}{1459713041} a^{10} - \frac{379823704}{1459713041} a^{9} + \frac{493709918}{1459713041} a^{8} + \frac{230256856}{1459713041} a^{7} - \frac{1268264551}{4379139123} a^{6} + \frac{160458095}{4379139123} a^{5} + \frac{746923456}{4379139123} a^{4} - \frac{934160479}{4379139123} a^{3} - \frac{270645920}{4379139123} a^{2} + \frac{754334224}{4379139123} a + \frac{1554696913}{4379139123}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $9$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{3778880}{270869} a^{19} + \frac{11227566}{270869} a^{18} - \frac{1330507}{270869} a^{17} - \frac{6019962}{270869} a^{16} - \frac{29575666}{270869} a^{15} + \frac{83845502}{270869} a^{14} - \frac{99234599}{270869} a^{13} - \frac{28701735}{270869} a^{12} - \frac{57842432}{270869} a^{11} + \frac{495643623}{270869} a^{10} - \frac{479848985}{270869} a^{9} + \frac{109683743}{270869} a^{8} - \frac{126499126}{270869} a^{7} + \frac{696261730}{270869} a^{6} - \frac{967858891}{270869} a^{5} + \frac{644490779}{270869} a^{4} - \frac{358982021}{270869} a^{3} + \frac{241607430}{270869} a^{2} - \frac{112651583}{270869} a + \frac{21204039}{270869}$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$99.9526828541$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$D_5\wr C_2$ (as 20T48):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 200 The 14 conjugacy class representatives for $D_5\wr C_2$ Character table for $D_5\wr C_2$

## 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 25 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5} 2.10.10.7x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
761Data not computed