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Properties

Label 20.8.50980770312...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{5}$
Root discriminant $54.37$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6029, 0, -2946, 0, -48653, 0, -27874, 0, 37645, 0, 14160, 0, -5199, 0, -2362, 0, -161, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 - 161*x^16 - 2362*x^14 - 5199*x^12 + 14160*x^10 + 37645*x^8 - 27874*x^6 - 48653*x^4 - 2946*x^2 + 6029)
 
gp: K = bnfinit(x^20 + 15*x^18 - 161*x^16 - 2362*x^14 - 5199*x^12 + 14160*x^10 + 37645*x^8 - 27874*x^6 - 48653*x^4 - 2946*x^2 + 6029, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} - 161 x^{16} - 2362 x^{14} - 5199 x^{12} + 14160 x^{10} + 37645 x^{8} - 27874 x^{6} - 48653 x^{4} - 2946 x^{2} + 6029 \)

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:  \(50980770312185223353600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.37$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{12} - \frac{3}{7} a^{10} + \frac{1}{7} a^{4} + \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{13} - \frac{3}{7} a^{11} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} + \frac{3}{7} a$, $\frac{1}{119} a^{16} - \frac{2}{119} a^{14} - \frac{45}{119} a^{12} + \frac{1}{17} a^{10} - \frac{7}{17} a^{8} + \frac{29}{119} a^{6} + \frac{52}{119} a^{4} + \frac{38}{119} a^{2} + \frac{7}{17}$, $\frac{1}{119} a^{17} - \frac{2}{119} a^{15} - \frac{45}{119} a^{13} + \frac{1}{17} a^{11} - \frac{7}{17} a^{9} + \frac{29}{119} a^{7} + \frac{52}{119} a^{5} + \frac{38}{119} a^{3} + \frac{7}{17} a$, $\frac{1}{9324224996767539449633} a^{18} + \frac{24248632501638220268}{9324224996767539449633} a^{16} - \frac{403247231880837449682}{9324224996767539449633} a^{14} + \frac{403915175376110942618}{1332032142395362778519} a^{12} - \frac{216212942044697455091}{1332032142395362778519} a^{10} + \frac{155150730462551900683}{548483823339267026449} a^{8} - \frac{774010549310577938132}{9324224996767539449633} a^{6} - \frac{138017738528097627406}{9324224996767539449633} a^{4} + \frac{373072439637813810798}{1332032142395362778519} a^{2} + \frac{148881732819369319935}{1332032142395362778519}$, $\frac{1}{9324224996767539449633} a^{19} + \frac{24248632501638220268}{9324224996767539449633} a^{17} - \frac{403247231880837449682}{9324224996767539449633} a^{15} + \frac{403915175376110942618}{1332032142395362778519} a^{13} - \frac{216212942044697455091}{1332032142395362778519} a^{11} + \frac{155150730462551900683}{548483823339267026449} a^{9} - \frac{774010549310577938132}{9324224996767539449633} a^{7} - \frac{138017738528097627406}{9324224996767539449633} a^{5} + \frac{373072439637813810798}{1332032142395362778519} a^{3} + \frac{148881732819369319935}{1332032142395362778519} a$

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:  \( 8028101172.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
6029Data not computed