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Properties

Label 20.0.21822182114...9209.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 883^{8}$
Root discriminant $26.12$
Ramified primes $3, 883$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -30, 0, 923, 0, 666, 0, 881, 0, 205, 0, 358, 0, 142, 0, 52, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 52*x^16 + 142*x^14 + 358*x^12 + 205*x^10 + 881*x^8 + 666*x^6 + 923*x^4 - 30*x^2 + 1)
 
gp: K = bnfinit(x^20 + 8*x^18 + 52*x^16 + 142*x^14 + 358*x^12 + 205*x^10 + 881*x^8 + 666*x^6 + 923*x^4 - 30*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} + 52 x^{16} + 142 x^{14} + 358 x^{12} + 205 x^{10} + 881 x^{8} + 666 x^{6} + 923 x^{4} - 30 x^{2} + 1 \)

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:  \(21822182114965484993955979209=3^{10}\cdot 883^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $3, 883$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{12} - \frac{1}{6} a^{10} - \frac{2}{9} a^{8} - \frac{1}{2} a^{7} - \frac{7}{18} a^{6} - \frac{1}{2} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{2} - \frac{5}{18}$, $\frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{2}{9} a^{9} - \frac{1}{2} a^{8} - \frac{7}{18} a^{7} - \frac{1}{2} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{3} - \frac{5}{18} a$, $\frac{1}{18} a^{14} - \frac{2}{9} a^{10} - \frac{1}{2} a^{9} - \frac{1}{18} a^{8} + \frac{1}{18} a^{6} - \frac{7}{18} a^{4} - \frac{4}{9} a^{2} - \frac{1}{3}$, $\frac{1}{18} a^{15} - \frac{2}{9} a^{11} - \frac{1}{18} a^{9} + \frac{1}{18} a^{7} + \frac{1}{9} a^{5} - \frac{1}{2} a^{4} - \frac{4}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{2}{9} a^{10} + \frac{1}{6} a^{8} - \frac{4}{9} a^{6} - \frac{1}{18} a^{4} - \frac{1}{2} a^{3} - \frac{1}{18} a^{2} - \frac{1}{2} a + \frac{7}{18}$, $\frac{1}{18} a^{17} - \frac{2}{9} a^{11} + \frac{1}{6} a^{9} - \frac{4}{9} a^{7} - \frac{1}{18} a^{5} - \frac{1}{2} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2} a^{2} + \frac{7}{18} a$, $\frac{1}{20230056630} a^{18} + \frac{19685981}{2890008090} a^{16} - \frac{110792701}{4046011326} a^{14} - \frac{18535736}{1123892035} a^{12} - \frac{190201483}{1123892035} a^{10} - \frac{1}{2} a^{9} - \frac{405114029}{1445004045} a^{8} + \frac{644774336}{10115028315} a^{6} + \frac{247081048}{1123892035} a^{4} - \frac{3013191331}{20230056630} a^{2} - \frac{1}{2} a - \frac{453038073}{1123892035}$, $\frac{1}{20230056630} a^{19} + \frac{19685981}{2890008090} a^{17} - \frac{110792701}{4046011326} a^{15} - \frac{18535736}{1123892035} a^{13} - \frac{190201483}{1123892035} a^{11} - \frac{405114029}{1445004045} a^{9} + \frac{644774336}{10115028315} a^{7} - \frac{629729939}{2247784070} a^{5} - \frac{1}{2} a^{4} - \frac{3013191331}{20230056630} a^{3} - \frac{453038073}{1123892035} a - \frac{1}{2}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{110304584}{3371676105} a^{18} - \frac{126285964}{481668015} a^{16} - \frac{1148937380}{674335221} a^{14} - \frac{15718123318}{3371676105} a^{12} - \frac{39546875749}{3371676105} a^{10} - \frac{3262541428}{481668015} a^{8} - \frac{32274665006}{1123892035} a^{6} - \frac{75903352451}{3371676105} a^{4} - \frac{101160878846}{3371676105} a^{2} + \frac{3287992976}{3371676105} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1357302.39442 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.7017201.1, 10.0.147723329623203.1, 10.4.147723329623203.2, 10.6.49241109874401.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
883Data not computed