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Properties

Label 20.8.63648839595...1904.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $49.00$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-439, -1464, 14947, -27164, -4859, 72180, -101662, 74738, -43527, 33292, -25957, 12906, -4213, 2382, -1426, 260, -11, 48, -11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 11*x^18 + 48*x^17 - 11*x^16 + 260*x^15 - 1426*x^14 + 2382*x^13 - 4213*x^12 + 12906*x^11 - 25957*x^10 + 33292*x^9 - 43527*x^8 + 74738*x^7 - 101662*x^6 + 72180*x^5 - 4859*x^4 - 27164*x^3 + 14947*x^2 - 1464*x - 439)
 
gp: K = bnfinit(x^20 - 4*x^19 - 11*x^18 + 48*x^17 - 11*x^16 + 260*x^15 - 1426*x^14 + 2382*x^13 - 4213*x^12 + 12906*x^11 - 25957*x^10 + 33292*x^9 - 43527*x^8 + 74738*x^7 - 101662*x^6 + 72180*x^5 - 4859*x^4 - 27164*x^3 + 14947*x^2 - 1464*x - 439, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 11 x^{18} + 48 x^{17} - 11 x^{16} + 260 x^{15} - 1426 x^{14} + 2382 x^{13} - 4213 x^{12} + 12906 x^{11} - 25957 x^{10} + 33292 x^{9} - 43527 x^{8} + 74738 x^{7} - 101662 x^{6} + 72180 x^{5} - 4859 x^{4} - 27164 x^{3} + 14947 x^{2} - 1464 x - 439 \)

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:  \(6364883959565411703183364041211904=2^{20}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.00$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{14} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{18} - \frac{2}{25} a^{17} + \frac{1}{25} a^{16} - \frac{2}{25} a^{15} + \frac{1}{25} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{8}{25} a^{11} - \frac{9}{25} a^{10} + \frac{1}{5} a^{9} + \frac{9}{25} a^{8} - \frac{1}{5} a^{7} - \frac{8}{25} a^{6} + \frac{7}{25} a^{5} + \frac{4}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{9}{25} a - \frac{11}{25}$, $\frac{1}{18056784338271362633648957584740700025} a^{19} - \frac{360854568298809280451997246587548131}{18056784338271362633648957584740700025} a^{18} - \frac{1509447091006067289820132102227241351}{18056784338271362633648957584740700025} a^{17} + \frac{1650936686960438141571865930919056194}{18056784338271362633648957584740700025} a^{16} + \frac{302514625742971583639717648240684109}{18056784338271362633648957584740700025} a^{15} + \frac{2680762568158074254817716555919371351}{18056784338271362633648957584740700025} a^{14} - \frac{1454133880130407533586563730269409176}{3611356867654272526729791516948140005} a^{13} - \frac{5860840485972425765662748220961586178}{18056784338271362633648957584740700025} a^{12} + \frac{5845023681612672586269285743472603433}{18056784338271362633648957584740700025} a^{11} + \frac{3156587209531008070260230859773766051}{18056784338271362633648957584740700025} a^{10} + \frac{3306444083230725720504588111600091839}{18056784338271362633648957584740700025} a^{9} + \frac{1625479667443748275029046097174366769}{18056784338271362633648957584740700025} a^{8} - \frac{5533561069404226238313371052231387163}{18056784338271362633648957584740700025} a^{7} + \frac{6751815725383300153257371137942339914}{18056784338271362633648957584740700025} a^{6} + \frac{5653299768913718635625170268083524366}{18056784338271362633648957584740700025} a^{5} + \frac{7016471673247200491164687341762467634}{18056784338271362633648957584740700025} a^{4} + \frac{1515377551665606067434195528789248411}{3611356867654272526729791516948140005} a^{3} + \frac{972366471545581885457104013140001271}{18056784338271362633648957584740700025} a^{2} - \frac{1335645556313708231027199106077189317}{3611356867654272526729791516948140005} a - \frac{896153940624723253833002234354590866}{18056784338271362633648957584740700025}$

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:  \( 3846298008.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$