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Properties

Label 20.0.54168219170...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $68.66$
Ramified primes $2, 3, 5, 23, 89$
Class number $31584$ (GRH)
Class group $[2, 2, 2, 3948]$ (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![595125, 0, 4786875, 0, 12243450, 0, 14430600, 0, 8749220, 0, 2829085, 0, 508110, 0, 51540, 0, 2910, 0, 85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 85*x^18 + 2910*x^16 + 51540*x^14 + 508110*x^12 + 2829085*x^10 + 8749220*x^8 + 14430600*x^6 + 12243450*x^4 + 4786875*x^2 + 595125)
 
gp: K = bnfinit(x^20 + 85*x^18 + 2910*x^16 + 51540*x^14 + 508110*x^12 + 2829085*x^10 + 8749220*x^8 + 14430600*x^6 + 12243450*x^4 + 4786875*x^2 + 595125, 1)
 

Normalized defining polynomial

\( x^{20} + 85 x^{18} + 2910 x^{16} + 51540 x^{14} + 508110 x^{12} + 2829085 x^{10} + 8749220 x^{8} + 14430600 x^{6} + 12243450 x^{4} + 4786875 x^{2} + 595125 \)

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:  \(5416821917044323616800000000000000000=2^{20}\cdot 3^{6}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.66$
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:  $2, 3, 5, 23, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{75} a^{16} - \frac{1}{15} a^{14} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{15} a^{6} + \frac{4}{15} a^{4}$, $\frac{1}{75} a^{17} - \frac{1}{15} a^{15} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{15} a^{7} + \frac{4}{15} a^{5}$, $\frac{1}{234940622881564667778075} a^{18} + \frac{66457987105808811086}{46988124576312933555615} a^{16} - \frac{210216545882578465833}{5220902730701437061735} a^{14} + \frac{915132062324800318853}{15662708192104311185205} a^{12} + \frac{5260169627505029906239}{15662708192104311185205} a^{10} + \frac{13876502481434700984722}{46988124576312933555615} a^{8} + \frac{9153923047726587531229}{46988124576312933555615} a^{6} + \frac{703952418852899136824}{3132541638420862237041} a^{4} - \frac{177159053480237353081}{3132541638420862237041} a^{2} - \frac{10158113130989094937}{45399154180012496189}$, $\frac{1}{234940622881564667778075} a^{19} + \frac{66457987105808811086}{46988124576312933555615} a^{17} - \frac{210216545882578465833}{5220902730701437061735} a^{15} + \frac{915132062324800318853}{15662708192104311185205} a^{13} + \frac{5260169627505029906239}{15662708192104311185205} a^{11} + \frac{13876502481434700984722}{46988124576312933555615} a^{9} + \frac{9153923047726587531229}{46988124576312933555615} a^{7} + \frac{703952418852899136824}{3132541638420862237041} a^{5} - \frac{177159053480237353081}{3132541638420862237041} a^{3} - \frac{10158113130989094937}{45399154180012496189} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{3948}$, which has order $31584$ (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:  \( -1 \) (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:  \( 366014.001413 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed