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Properties

Label 20.8.15583578925...2688.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 3^{5}\cdot 11^{19}$
Root discriminant $25.68$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, 636, 2214, 3274, 1226, -1898, -2809, -2428, -1069, -76, -286, 144, 262, -52, 106, 82, -6, 18, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 + 18*x^17 - 6*x^16 + 82*x^15 + 106*x^14 - 52*x^13 + 262*x^12 + 144*x^11 - 286*x^10 - 76*x^9 - 1069*x^8 - 2428*x^7 - 2809*x^6 - 1898*x^5 + 1226*x^4 + 3274*x^3 + 2214*x^2 + 636*x + 67)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 + 18*x^17 - 6*x^16 + 82*x^15 + 106*x^14 - 52*x^13 + 262*x^12 + 144*x^11 - 286*x^10 - 76*x^9 - 1069*x^8 - 2428*x^7 - 2809*x^6 - 1898*x^5 + 1226*x^4 + 3274*x^3 + 2214*x^2 + 636*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} + 18 x^{17} - 6 x^{16} + 82 x^{15} + 106 x^{14} - 52 x^{13} + 262 x^{12} + 144 x^{11} - 286 x^{10} - 76 x^{9} - 1069 x^{8} - 2428 x^{7} - 2809 x^{6} - 1898 x^{5} + 1226 x^{4} + 3274 x^{3} + 2214 x^{2} + 636 x + 67 \)

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:  \(15583578925526925703866482688=2^{20}\cdot 3^{5}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3244846248071731596929325386621} a^{19} + \frac{945112213152868168141401249272}{3244846248071731596929325386621} a^{18} - \frac{105756626131281920405443235150}{3244846248071731596929325386621} a^{17} - \frac{1045433371091852021832621411428}{3244846248071731596929325386621} a^{16} - \frac{1070353424579918467399143536279}{3244846248071731596929325386621} a^{15} + \frac{1258368548104269519955185377242}{3244846248071731596929325386621} a^{14} + \frac{1574505397726672225640964948063}{3244846248071731596929325386621} a^{13} + \frac{1333216819320396362055062354859}{3244846248071731596929325386621} a^{12} - \frac{654195758624412810064108106440}{3244846248071731596929325386621} a^{11} + \frac{527779916235247727895745435855}{3244846248071731596929325386621} a^{10} + \frac{1327745326331353218083182782262}{3244846248071731596929325386621} a^{9} - \frac{107397194950242946239735372074}{3244846248071731596929325386621} a^{8} - \frac{1115790395631960951751322150645}{3244846248071731596929325386621} a^{7} - \frac{719494301938058078884768975871}{3244846248071731596929325386621} a^{6} + \frac{8419164408971415291330419985}{3244846248071731596929325386621} a^{5} + \frac{1369132841726269487614132341884}{3244846248071731596929325386621} a^{4} - \frac{1515897431923256686482130622258}{3244846248071731596929325386621} a^{3} + \frac{1521546033661342173254324536584}{3244846248071731596929325386621} a^{2} + \frac{544636790216164337841566593023}{3244846248071731596929325386621} a - \frac{1320739759568768594745052752756}{3244846248071731596929325386621}$

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:  \( 3040136.91068 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.2414538435584.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed