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Properties

Label 20.12.1928006729...3024.1
Degree $20$
Signature $[12, 4]$
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 20T335

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7369, 0, -29476, 0, 25127, 0, 13509, 0, -15094, 0, 233, 0, 1718, 0, -131, 0, -69, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - 69*x^16 - 131*x^14 + 1718*x^12 + 233*x^10 - 15094*x^8 + 13509*x^6 + 25127*x^4 - 29476*x^2 + 7369)
 
gp: K = bnfinit(x^20 + 4*x^18 - 69*x^16 - 131*x^14 + 1718*x^12 + 233*x^10 - 15094*x^8 + 13509*x^6 + 25127*x^4 - 29476*x^2 + 7369, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - 69 x^{16} - 131 x^{14} + 1718 x^{12} + 233 x^{10} - 15094 x^{8} + 13509 x^{6} + 25127 x^{4} - 29476 x^{2} + 7369 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6752094923858628917} a^{18} + \frac{1040294612983957602}{6752094923858628917} a^{16} - \frac{1428870185125686116}{6752094923858628917} a^{14} + \frac{1841322116954007957}{6752094923858628917} a^{12} + \frac{44952955711912532}{6752094923858628917} a^{10} + \frac{2648237959440869815}{6752094923858628917} a^{8} + \frac{1965471790451264302}{6752094923858628917} a^{6} - \frac{2615630748079801143}{6752094923858628917} a^{4} - \frac{370901698641516382}{6752094923858628917} a^{2} + \frac{776529879808063373}{6752094923858628917}$, $\frac{1}{6752094923858628917} a^{19} + \frac{1040294612983957602}{6752094923858628917} a^{17} - \frac{1428870185125686116}{6752094923858628917} a^{15} + \frac{1841322116954007957}{6752094923858628917} a^{13} + \frac{44952955711912532}{6752094923858628917} a^{11} + \frac{2648237959440869815}{6752094923858628917} a^{9} + \frac{1965471790451264302}{6752094923858628917} a^{7} - \frac{2615630748079801143}{6752094923858628917} a^{5} - \frac{370901698641516382}{6752094923858628917} a^{3} + \frac{776529879808063373}{6752094923858628917} a$

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:  $15$
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:  \( 7296280492.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T335:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n335 are not computed
Character table for t20n335 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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
7369Data not computed