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Properties

Label 20.20.5597247761...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{15}\cdot 211^{10}$
Root discriminant $97.14$
Ramified primes $2, 5, 211$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T534

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139128125, 0, -532775000, 0, 683833125, 0, -407954375, 0, 133431125, 0, -25767125, 0, 3000875, 0, -206525, 0, 7860, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 145*x^18 + 7860*x^16 - 206525*x^14 + 3000875*x^12 - 25767125*x^10 + 133431125*x^8 - 407954375*x^6 + 683833125*x^4 - 532775000*x^2 + 139128125)
 
gp: K = bnfinit(x^20 - 145*x^18 + 7860*x^16 - 206525*x^14 + 3000875*x^12 - 25767125*x^10 + 133431125*x^8 - 407954375*x^6 + 683833125*x^4 - 532775000*x^2 + 139128125, 1)
 

Normalized defining polynomial

\( x^{20} - 145 x^{18} + 7860 x^{16} - 206525 x^{14} + 3000875 x^{12} - 25767125 x^{10} + 133431125 x^{8} - 407954375 x^{6} + 683833125 x^{4} - 532775000 x^{2} + 139128125 \)

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:  \(5597247761133055315411232000000000000000=2^{20}\cdot 5^{15}\cdot 211^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.14$
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:  $2, 5, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{6407894308041337327368125} a^{18} - \frac{3020498496816058240792}{6407894308041337327368125} a^{16} + \frac{1238665773658332448823}{1281578861608267465473625} a^{14} - \frac{1978954719081468449569}{1281578861608267465473625} a^{12} - \frac{1683798072562777105624}{256315772321653493094725} a^{10} - \frac{2813838522058287392719}{256315772321653493094725} a^{8} + \frac{387891878859728251}{48590667738702083999} a^{6} - \frac{1877428183073469646373}{51263154464330698618945} a^{4} - \frac{2828889038935596373540}{10252630892866139723789} a^{2} - \frac{16596330024407840607}{48590667738702083999}$, $\frac{1}{6407894308041337327368125} a^{19} - \frac{3020498496816058240792}{6407894308041337327368125} a^{17} + \frac{1238665773658332448823}{1281578861608267465473625} a^{15} - \frac{1978954719081468449569}{1281578861608267465473625} a^{13} - \frac{1683798072562777105624}{256315772321653493094725} a^{11} - \frac{2813838522058287392719}{256315772321653493094725} a^{9} + \frac{387891878859728251}{48590667738702083999} a^{7} - \frac{1877428183073469646373}{51263154464330698618945} a^{5} - \frac{2828889038935596373540}{10252630892866139723789} a^{3} - \frac{16596330024407840607}{48590667738702083999} a$

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:  \( 25618243093000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T534:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n534 are not computed
Character table for t20n534 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.6194123253125.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
5Data not computed
211Data not computed