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Properties

Label 20.0.55972477611...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 211^{10}$
Root discriminant $97.14$
Ramified primes $2, 5, 211$
Class number $649152$ (GRH)
Class group $[2, 2, 162288]$ (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, 686409375, 0, 983721875, 0, 632772500, 0, 211101500, 0, 38655750, 0, 3998975, 0, 237375, 0, 7965, 0, 140, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 140*x^18 + 7965*x^16 + 237375*x^14 + 3998975*x^12 + 38655750*x^10 + 211101500*x^8 + 632772500*x^6 + 983721875*x^4 + 686409375*x^2 + 139128125)
 
gp: K = bnfinit(x^20 + 140*x^18 + 7965*x^16 + 237375*x^14 + 3998975*x^12 + 38655750*x^10 + 211101500*x^8 + 632772500*x^6 + 983721875*x^4 + 686409375*x^2 + 139128125, 1)
 

Normalized defining polynomial

\( x^{20} + 140 x^{18} + 7965 x^{16} + 237375 x^{14} + 3998975 x^{12} + 38655750 x^{10} + 211101500 x^{8} + 632772500 x^{6} + 983721875 x^{4} + 686409375 x^{2} + 139128125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{12457193088588845788092074375} a^{18} + \frac{8720464879222755382965457}{12457193088588845788092074375} a^{16} - \frac{98653454482765748308464}{99657544708710766304736595} a^{14} + \frac{19401489712638477490831}{11807765960747721126153625} a^{12} - \frac{5332156010401744323861888}{498287723543553831523682975} a^{10} + \frac{9690908050044573500809269}{498287723543553831523682975} a^{8} - \frac{5182027956211167812363512}{99657544708710766304736595} a^{6} + \frac{6134065379969590272679751}{99657544708710766304736595} a^{4} - \frac{4589786200790349549336564}{19931508941742153260947319} a^{2} + \frac{37233077439970272300316}{94462127685981769009229}$, $\frac{1}{12457193088588845788092074375} a^{19} + \frac{8720464879222755382965457}{12457193088588845788092074375} a^{17} - \frac{98653454482765748308464}{99657544708710766304736595} a^{15} + \frac{19401489712638477490831}{11807765960747721126153625} a^{13} - \frac{5332156010401744323861888}{498287723543553831523682975} a^{11} + \frac{9690908050044573500809269}{498287723543553831523682975} a^{9} - \frac{5182027956211167812363512}{99657544708710766304736595} a^{7} + \frac{6134065379969590272679751}{99657544708710766304736595} a^{5} - \frac{4589786200790349549336564}{19931508941742153260947319} a^{3} + \frac{37233077439970272300316}{94462127685981769009229} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{162288}$, which has order $649152$ (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:  $9$
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:  \( 454775.127492 \) (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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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