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Properties

Label 20.8.30517578125...0000.9
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{35}$
Root discriminant $33.44$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T344

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, 625, 0, 625, 0, -375, 0, -625, 0, 50, 0, 200, 0, 0, 0, -25, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^16 + 200*x^12 + 50*x^10 - 625*x^8 - 375*x^6 + 625*x^4 + 625*x^2 + 125)
 
gp: K = bnfinit(x^20 - 25*x^16 + 200*x^12 + 50*x^10 - 625*x^8 - 375*x^6 + 625*x^4 + 625*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{16} + 200 x^{12} + 50 x^{10} - 625 x^{8} - 375 x^{6} + 625 x^{4} + 625 x^{2} + 125 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3051757812500000000000000000000=2^{20}\cdot 5^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.44$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{25} a^{14}$, $\frac{1}{25} a^{15}$, $\frac{1}{25} a^{16}$, $\frac{1}{25} a^{17}$, $\frac{1}{2525} a^{18} - \frac{28}{2525} a^{16} - \frac{49}{2525} a^{14} + \frac{32}{505} a^{12} - \frac{48}{505} a^{10} + \frac{41}{505} a^{8} + \frac{8}{101} a^{6} - \frac{37}{101} a^{4} - \frac{50}{101} a^{2} + \frac{11}{101}$, $\frac{1}{2525} a^{19} - \frac{28}{2525} a^{17} - \frac{49}{2525} a^{15} + \frac{32}{505} a^{13} - \frac{48}{505} a^{11} + \frac{41}{505} a^{9} + \frac{8}{101} a^{7} - \frac{37}{101} a^{5} - \frac{50}{101} a^{3} + \frac{11}{101} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 37047306.6062 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T344:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n344 are not computed
Character table for t20n344 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed