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Properties

Label 20.6.84961403747...0000.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{34}\cdot 5^{12}\cdot 1193^{4}$
Root discriminant $35.19$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49, 32, 818, -1748, -1009, 5760, -3876, -3182, 4933, -1748, 32, 576, -1254, 878, -235, 4, 40, -38, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 38*x^17 + 40*x^16 + 4*x^15 - 235*x^14 + 878*x^13 - 1254*x^12 + 576*x^11 + 32*x^10 - 1748*x^9 + 4933*x^8 - 3182*x^7 - 3876*x^6 + 5760*x^5 - 1009*x^4 - 1748*x^3 + 818*x^2 + 32*x - 49)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 38*x^17 + 40*x^16 + 4*x^15 - 235*x^14 + 878*x^13 - 1254*x^12 + 576*x^11 + 32*x^10 - 1748*x^9 + 4933*x^8 - 3182*x^7 - 3876*x^6 + 5760*x^5 - 1009*x^4 - 1748*x^3 + 818*x^2 + 32*x - 49, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 38 x^{17} + 40 x^{16} + 4 x^{15} - 235 x^{14} + 878 x^{13} - 1254 x^{12} + 576 x^{11} + 32 x^{10} - 1748 x^{9} + 4933 x^{8} - 3182 x^{7} - 3876 x^{6} + 5760 x^{5} - 1009 x^{4} - 1748 x^{3} + 818 x^{2} + 32 x - 49 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8496140374773858304000000000000=-\,2^{34}\cdot 5^{12}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.19$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2619954348418257919930852235715665} a^{19} - \frac{207050851679130596768863184784328}{523990869683651583986170447143133} a^{18} - \frac{97255094638492161635116451899843}{201534949878327532302373248901205} a^{17} + \frac{457177608271542123908852671313618}{2619954348418257919930852235715665} a^{16} + \frac{487436999485998885142930720052383}{2619954348418257919930852235715665} a^{15} - \frac{72510707094312883549722370719088}{2619954348418257919930852235715665} a^{14} - \frac{1131762902830305474246248279019363}{2619954348418257919930852235715665} a^{13} + \frac{80041874170429732306641205913488}{523990869683651583986170447143133} a^{12} + \frac{147740791971517584270585552694226}{2619954348418257919930852235715665} a^{11} - \frac{285736600349041663752615369450353}{2619954348418257919930852235715665} a^{10} - \frac{1280185718486477546338875672433931}{2619954348418257919930852235715665} a^{9} + \frac{5935475415251382828509462794802}{49433100913552036225110419541805} a^{8} - \frac{131682656289777120252804186197841}{2619954348418257919930852235715665} a^{7} - \frac{1107848728117716120688790755529943}{2619954348418257919930852235715665} a^{6} - \frac{123384024597873180858284274617489}{2619954348418257919930852235715665} a^{5} - \frac{261162644760040145368872268183724}{2619954348418257919930852235715665} a^{4} + \frac{200644405459459728631365375372837}{2619954348418257919930852235715665} a^{3} + \frac{410047161794235248218344366633154}{2619954348418257919930852235715665} a^{2} - \frac{95842747278968734290509433574868}{2619954348418257919930852235715665} a - \frac{482400729535804701873116598777126}{2619954348418257919930852235715665}$

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:  $12$
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:  \( 111883328.474 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
1193Data not computed