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Properties

Label 20.0.31025647067...2368.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 79^{10}$
Root discriminant $14.95$
Ramified primes $2, 79$
Class number $1$
Class group Trivial
Galois group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, 48, 0, -24, 0, -40, 0, 50, 0, -3, 0, 25, 0, -10, 0, -3, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 3*x^16 - 10*x^14 + 25*x^12 - 3*x^10 + 50*x^8 - 40*x^6 - 24*x^4 + 48*x^2 + 32)
 
gp: K = bnfinit(x^20 + 3*x^18 - 3*x^16 - 10*x^14 + 25*x^12 - 3*x^10 + 50*x^8 - 40*x^6 - 24*x^4 + 48*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 3 x^{16} - 10 x^{14} + 25 x^{12} - 3 x^{10} + 50 x^{8} - 40 x^{6} - 24 x^{4} + 48 x^{2} + 32 \)

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:  \(310256470675516529082368=2^{15}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.95$
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:  $2, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{5}{18} a^{4} + \frac{5}{18} a^{2} - \frac{2}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{5}{18} a^{5} + \frac{5}{18} a^{3} - \frac{2}{9} a$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{12} - \frac{1}{36} a^{10} + \frac{13}{36} a^{6} + \frac{5}{36} a^{4} - \frac{1}{2} a^{3} + \frac{7}{18} a^{2}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{5}{36} a^{7} - \frac{1}{2} a^{6} - \frac{13}{36} a^{5} - \frac{1}{2} a^{4} - \frac{1}{9} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{648} a^{16} + \frac{7}{648} a^{14} + \frac{5}{648} a^{12} - \frac{11}{324} a^{10} + \frac{53}{648} a^{8} + \frac{217}{648} a^{6} - \frac{1}{2} a^{5} + \frac{91}{324} a^{4} - \frac{1}{2} a^{3} - \frac{38}{81} a^{2} - \frac{16}{81}$, $\frac{1}{648} a^{17} + \frac{7}{648} a^{15} + \frac{5}{648} a^{13} - \frac{11}{324} a^{11} + \frac{53}{648} a^{9} - \frac{107}{648} a^{7} - \frac{71}{324} a^{5} - \frac{1}{2} a^{4} + \frac{5}{162} a^{3} - \frac{1}{2} a^{2} - \frac{16}{81} a$, $\frac{1}{9072} a^{18} - \frac{1}{9072} a^{16} + \frac{19}{3024} a^{14} + \frac{11}{648} a^{12} - \frac{311}{9072} a^{10} + \frac{1}{1008} a^{8} - \frac{73}{216} a^{6} - \frac{355}{1134} a^{4} + \frac{47}{126} a^{2} - \frac{206}{567}$, $\frac{1}{9072} a^{19} - \frac{1}{9072} a^{17} + \frac{19}{3024} a^{15} + \frac{11}{648} a^{13} - \frac{311}{9072} a^{11} + \frac{1}{1008} a^{9} + \frac{35}{216} a^{7} - \frac{1}{2} a^{6} + \frac{106}{567} a^{5} - \frac{8}{63} a^{3} - \frac{1}{2} a^{2} - \frac{206}{567} a$

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:  \( 2246.68851207 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{-79}) \), 4.0.49928.1, 5.1.6241.1 x5, 10.0.3077056399.1

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$