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Properties

Label 20.20.1928006729...3024.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{16}\cdot 7369^{3}$
Root discriminant $51.79$
Ramified primes $2, 11, 7369$
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![7369, 0, -187263, 0, 600924, 0, -803006, 0, 575732, 0, -244682, 0, 64059, 0, -10331, 0, 986, 0, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 50*x^18 + 986*x^16 - 10331*x^14 + 64059*x^12 - 244682*x^10 + 575732*x^8 - 803006*x^6 + 600924*x^4 - 187263*x^2 + 7369)
 
gp: K = bnfinit(x^20 - 50*x^18 + 986*x^16 - 10331*x^14 + 64059*x^12 - 244682*x^10 + 575732*x^8 - 803006*x^6 + 600924*x^4 - 187263*x^2 + 7369, 1)
 

Normalized defining polynomial

\( x^{20} - 50 x^{18} + 986 x^{16} - 10331 x^{14} + 64059 x^{12} - 244682 x^{10} + 575732 x^{8} - 803006 x^{6} + 600924 x^{4} - 187263 x^{2} + 7369 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19280067292868642604792128836993024=2^{20}\cdot 11^{16}\cdot 7369^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.79$
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:  $2, 11, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{86010465950649899} a^{18} - \frac{7323410031133908}{86010465950649899} a^{16} + \frac{42870031595130773}{86010465950649899} a^{14} + \frac{40559295939175125}{86010465950649899} a^{12} + \frac{20725585435113740}{86010465950649899} a^{10} + \frac{10413391619388361}{86010465950649899} a^{8} - \frac{24170053912101355}{86010465950649899} a^{6} - \frac{2408250091122395}{86010465950649899} a^{4} + \frac{29847034409915666}{86010465950649899} a^{2} + \frac{2930460585206891}{86010465950649899}$, $\frac{1}{86010465950649899} a^{19} - \frac{7323410031133908}{86010465950649899} a^{17} + \frac{42870031595130773}{86010465950649899} a^{15} + \frac{40559295939175125}{86010465950649899} a^{13} + \frac{20725585435113740}{86010465950649899} a^{11} + \frac{10413391619388361}{86010465950649899} a^{9} - \frac{24170053912101355}{86010465950649899} a^{7} - \frac{2408250091122395}{86010465950649899} a^{5} + \frac{29847034409915666}{86010465950649899} a^{3} + \frac{2930460585206891}{86010465950649899} a$

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:  $19$
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:  \( 49257443499.3 \) (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

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
11Data not computed
7369Data not computed