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Properties

Label 20.0.23997407597...4112.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 7^{10}\cdot 11^{9}$
Root discriminant $52.36$
Ramified primes $2, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99431287648, 0, 28408939328, 0, 3043814928, 0, 289887136, 0, 41412448, 0, 2689120, 0, 220892, 0, 13720, 0, 882, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 28*x^18 + 882*x^16 + 13720*x^14 + 220892*x^12 + 2689120*x^10 + 41412448*x^8 + 289887136*x^6 + 3043814928*x^4 + 28408939328*x^2 + 99431287648)
 
gp: K = bnfinit(x^20 + 28*x^18 + 882*x^16 + 13720*x^14 + 220892*x^12 + 2689120*x^10 + 41412448*x^8 + 289887136*x^6 + 3043814928*x^4 + 28408939328*x^2 + 99431287648, 1)
 

Normalized defining polynomial

\( x^{20} + 28 x^{18} + 882 x^{16} + 13720 x^{14} + 220892 x^{12} + 2689120 x^{10} + 41412448 x^{8} + 289887136 x^{6} + 3043814928 x^{4} + 28408939328 x^{2} + 99431287648 \)

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:  \(23997407597237747473858076304474112=2^{55}\cdot 7^{10}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.36$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{98} a^{4}$, $\frac{1}{98} a^{5}$, $\frac{1}{686} a^{6}$, $\frac{1}{686} a^{7}$, $\frac{1}{9604} a^{8}$, $\frac{1}{9604} a^{9}$, $\frac{1}{67228} a^{10}$, $\frac{1}{67228} a^{11}$, $\frac{1}{941192} a^{12}$, $\frac{1}{941192} a^{13}$, $\frac{1}{6588344} a^{14}$, $\frac{1}{6588344} a^{15}$, $\frac{1}{92236816} a^{16}$, $\frac{1}{92236816} a^{17}$, $\frac{1}{12993435965567792} a^{18} - \frac{119677}{66293040640652} a^{16} - \frac{1251545}{16573260160163} a^{14} - \frac{854403}{9470434377236} a^{12} + \frac{203587}{676459598374} a^{10} + \frac{1633200}{48318542741} a^{8} - \frac{7047079}{13805297926} a^{6} - \frac{5489405}{1972185418} a^{4} + \frac{970885}{140870387} a^{2} + \frac{4788209}{20124341}$, $\frac{1}{12993435965567792} a^{19} - \frac{119677}{66293040640652} a^{17} - \frac{1251545}{16573260160163} a^{15} - \frac{854403}{9470434377236} a^{13} + \frac{203587}{676459598374} a^{11} + \frac{1633200}{48318542741} a^{9} - \frac{7047079}{13805297926} a^{7} - \frac{5489405}{1972185418} a^{5} + \frac{970885}{140870387} a^{3} + \frac{4788209}{20124341} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.1103872.6, 10.0.479756288.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
7Data not computed
$11$11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$