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Properties

Label 20.4.59506232715...7669.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 109^{3}$
Root discriminant $13.76$
Ramified primes $11, 109$
Class number $1$
Class group Trivial
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 3, -8, 16, -34, 39, -30, -3, 46, -57, 46, -3, -30, 39, -34, 16, -8, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 8*x^17 + 16*x^16 - 34*x^15 + 39*x^14 - 30*x^13 - 3*x^12 + 46*x^11 - 57*x^10 + 46*x^9 - 3*x^8 - 30*x^7 + 39*x^6 - 34*x^5 + 16*x^4 - 8*x^3 + 3*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 8*x^17 + 16*x^16 - 34*x^15 + 39*x^14 - 30*x^13 - 3*x^12 + 46*x^11 - 57*x^10 + 46*x^9 - 3*x^8 - 30*x^7 + 39*x^6 - 34*x^5 + 16*x^4 - 8*x^3 + 3*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} - 8 x^{17} + 16 x^{16} - 34 x^{15} + 39 x^{14} - 30 x^{13} - 3 x^{12} + 46 x^{11} - 57 x^{10} + 46 x^{9} - 3 x^{8} - 30 x^{7} + 39 x^{6} - 34 x^{5} + 16 x^{4} - 8 x^{3} + 3 x^{2} - x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59506232715491992087669=11^{16}\cdot 109^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.76$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8623} a^{18} - \frac{560}{8623} a^{17} + \frac{2614}{8623} a^{16} - \frac{3387}{8623} a^{15} + \frac{2298}{8623} a^{14} + \frac{3598}{8623} a^{13} + \frac{4241}{8623} a^{12} - \frac{3022}{8623} a^{11} + \frac{3569}{8623} a^{10} - \frac{90}{8623} a^{9} + \frac{3569}{8623} a^{8} - \frac{3022}{8623} a^{7} + \frac{4241}{8623} a^{6} + \frac{3598}{8623} a^{5} + \frac{2298}{8623} a^{4} - \frac{3387}{8623} a^{3} + \frac{2614}{8623} a^{2} - \frac{560}{8623} a + \frac{1}{8623}$, $\frac{1}{8623} a^{19} - \frac{558}{8623} a^{17} + \frac{3166}{8623} a^{16} + \frac{2638}{8623} a^{15} - \frac{2972}{8623} a^{14} + \frac{1339}{8623} a^{13} + \frac{613}{8623} a^{12} + \frac{1357}{8623} a^{11} - \frac{1986}{8623} a^{10} - \frac{3716}{8623} a^{9} + \frac{3705}{8623} a^{8} + \frac{2029}{8623} a^{7} - \frac{1390}{8623} a^{6} - \frac{604}{8623} a^{5} - \frac{1334}{8623} a^{4} + \frac{2954}{8623} a^{3} - \frac{2630}{8623} a^{2} - \frac{3171}{8623} a + \frac{560}{8623}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2017.51236698 \)
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