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Properties

Label 20.8.70699355089...5389.2
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 109^{5}$
Root discriminant $22.00$
Ramified primes $11, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131, -383, -72, 1391, -1747, 1175, -2688, 5854, -6583, 4324, -1883, 364, 592, -867, 483, -27, -113, 53, -1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - x^18 + 53*x^17 - 113*x^16 - 27*x^15 + 483*x^14 - 867*x^13 + 592*x^12 + 364*x^11 - 1883*x^10 + 4324*x^9 - 6583*x^8 + 5854*x^7 - 2688*x^6 + 1175*x^5 - 1747*x^4 + 1391*x^3 - 72*x^2 - 383*x + 131)
 
gp: K = bnfinit(x^20 - 5*x^19 - x^18 + 53*x^17 - 113*x^16 - 27*x^15 + 483*x^14 - 867*x^13 + 592*x^12 + 364*x^11 - 1883*x^10 + 4324*x^9 - 6583*x^8 + 5854*x^7 - 2688*x^6 + 1175*x^5 - 1747*x^4 + 1391*x^3 - 72*x^2 - 383*x + 131, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - x^{18} + 53 x^{17} - 113 x^{16} - 27 x^{15} + 483 x^{14} - 867 x^{13} + 592 x^{12} + 364 x^{11} - 1883 x^{10} + 4324 x^{9} - 6583 x^{8} + 5854 x^{7} - 2688 x^{6} + 1175 x^{5} - 1747 x^{4} + 1391 x^{3} - 72 x^{2} - 383 x + 131 \)

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:  \(706993550892760357993595389=11^{16}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.00$
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:  $11, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13681230928171670753} a^{19} - \frac{3097360628439901999}{13681230928171670753} a^{18} - \frac{3976014505879992350}{13681230928171670753} a^{17} - \frac{1163475380149501164}{13681230928171670753} a^{16} + \frac{2981793910499420992}{13681230928171670753} a^{15} - \frac{5881409690402677205}{13681230928171670753} a^{14} - \frac{818116281069161773}{13681230928171670753} a^{13} - \frac{5639256138902812389}{13681230928171670753} a^{12} + \frac{3844220971159417893}{13681230928171670753} a^{11} - \frac{2450397684105578260}{13681230928171670753} a^{10} + \frac{2603720107172197112}{13681230928171670753} a^{9} + \frac{5227227492332868398}{13681230928171670753} a^{8} - \frac{1962079629151407231}{13681230928171670753} a^{7} + \frac{4625300681575842413}{13681230928171670753} a^{6} - \frac{2856144250953203807}{13681230928171670753} a^{5} + \frac{1093969166445628512}{13681230928171670753} a^{4} - \frac{5035835080924670336}{13681230928171670753} a^{3} + \frac{1764478704006025753}{13681230928171670753} a^{2} + \frac{5577632054402430320}{13681230928171670753} a + \frac{39316983911067616}{104436877314287563}$

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:  \( 605693.43977 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.23365118029.1

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
Arithmetically equvalently 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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20$ $20$ $20$

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
11Data not computed
109Data not computed