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Properties

Label 20.10.1337308814...8256.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{44}\cdot 31^{5}\cdot 227^{4}$
Root discriminant $32.09$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 128, -532, -128, 2626, -2724, -3827, 6930, 1807, -4904, -717, 962, 897, -372, -289, 30, 53, 16, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 11*x^18 + 16*x^17 + 53*x^16 + 30*x^15 - 289*x^14 - 372*x^13 + 897*x^12 + 962*x^11 - 717*x^10 - 4904*x^9 + 1807*x^8 + 6930*x^7 - 3827*x^6 - 2724*x^5 + 2626*x^4 - 128*x^3 - 532*x^2 + 128*x - 8)
 
gp: K = bnfinit(x^20 - 2*x^19 - 11*x^18 + 16*x^17 + 53*x^16 + 30*x^15 - 289*x^14 - 372*x^13 + 897*x^12 + 962*x^11 - 717*x^10 - 4904*x^9 + 1807*x^8 + 6930*x^7 - 3827*x^6 - 2724*x^5 + 2626*x^4 - 128*x^3 - 532*x^2 + 128*x - 8, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 11 x^{18} + 16 x^{17} + 53 x^{16} + 30 x^{15} - 289 x^{14} - 372 x^{13} + 897 x^{12} + 962 x^{11} - 717 x^{10} - 4904 x^{9} + 1807 x^{8} + 6930 x^{7} - 3827 x^{6} - 2724 x^{5} + 2626 x^{4} - 128 x^{3} - 532 x^{2} + 128 x - 8 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1337308814535692506945873248256=-\,2^{44}\cdot 31^{5}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.09$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{18} + \frac{1}{16} a^{14} - \frac{1}{8} a^{12} + \frac{3}{16} a^{10} - \frac{1}{4} a^{8} - \frac{1}{16} a^{6} + \frac{3}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{130654212359852277475685536} a^{19} - \frac{1393954910343892977110137}{130654212359852277475685536} a^{18} + \frac{1348240313452701606382035}{65327106179926138737842768} a^{17} - \frac{1235632114889461381438481}{65327106179926138737842768} a^{16} - \frac{1576183807362432769095581}{130654212359852277475685536} a^{15} + \frac{2051255046744991614754593}{130654212359852277475685536} a^{14} + \frac{7732313037101230748724877}{65327106179926138737842768} a^{13} - \frac{6625347828251733270228981}{65327106179926138737842768} a^{12} + \frac{19841434688170573055859931}{130654212359852277475685536} a^{11} + \frac{10442022103694158314187301}{130654212359852277475685536} a^{10} - \frac{198478541339876370330661}{65327106179926138737842768} a^{9} + \frac{1399372316332972517546083}{65327106179926138737842768} a^{8} - \frac{9404910240814482188088627}{130654212359852277475685536} a^{7} - \frac{17603687284113614406271297}{130654212359852277475685536} a^{6} + \frac{4603300129856438561170633}{65327106179926138737842768} a^{5} + \frac{9022884636653027581218255}{65327106179926138737842768} a^{4} + \frac{15152095367116083306791557}{32663553089963069368921384} a^{3} - \frac{11744790405833882380797087}{32663553089963069368921384} a^{2} - \frac{5256341802116029294937993}{16331776544981534684460692} a - \frac{5240932229971736135731717}{16331776544981534684460692}$

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:  $14$
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:  \( 243482042.52 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
31Data not computed
227Data not computed