↑

Properties

Label 20.12.2729663703...4832.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{34}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $46.97$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![968, 25872, 68688, -119520, -351031, 171608, 603232, -152650, -413721, 127220, 145925, -63410, -27089, 17320, 2075, -2588, 82, 198, -23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 23*x^18 + 198*x^17 + 82*x^16 - 2588*x^15 + 2075*x^14 + 17320*x^13 - 27089*x^12 - 63410*x^11 + 145925*x^10 + 127220*x^9 - 413721*x^8 - 152650*x^7 + 603232*x^6 + 171608*x^5 - 351031*x^4 - 119520*x^3 + 68688*x^2 + 25872*x + 968)
 
gp: K = bnfinit(x^20 - 6*x^19 - 23*x^18 + 198*x^17 + 82*x^16 - 2588*x^15 + 2075*x^14 + 17320*x^13 - 27089*x^12 - 63410*x^11 + 145925*x^10 + 127220*x^9 - 413721*x^8 - 152650*x^7 + 603232*x^6 + 171608*x^5 - 351031*x^4 - 119520*x^3 + 68688*x^2 + 25872*x + 968, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 23 x^{18} + 198 x^{17} + 82 x^{16} - 2588 x^{15} + 2075 x^{14} + 17320 x^{13} - 27089 x^{12} - 63410 x^{11} + 145925 x^{10} + 127220 x^{9} - 413721 x^{8} - 152650 x^{7} + 603232 x^{6} + 171608 x^{5} - 351031 x^{4} - 119520 x^{3} + 68688 x^{2} + 25872 x + 968 \)

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:  \(2729663703817172900595808790904832=2^{34}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.97$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{12959236999397683492782133631770074387441052} a^{19} - \frac{44934841608264510626367053331016200460241}{3239809249849420873195533407942518596860263} a^{18} - \frac{1177905205887638345055222542831587490858515}{12959236999397683492782133631770074387441052} a^{17} - \frac{129712731210857111007956316921848233554293}{294528113622674624835957582540228963350933} a^{16} - \frac{2754600588227484477708247202829077218001151}{6479618499698841746391066815885037193720526} a^{15} + \frac{862444503528529711618592260983712250436381}{3239809249849420873195533407942518596860263} a^{14} - \frac{4551701766023123023799620872900290957796401}{12959236999397683492782133631770074387441052} a^{13} + \frac{65901707312629217680990047758830185371981}{6479618499698841746391066815885037193720526} a^{12} - \frac{5807115615743629389728287486697777661439161}{12959236999397683492782133631770074387441052} a^{11} - \frac{764377690913968233464106947168496839644980}{3239809249849420873195533407942518596860263} a^{10} - \frac{3397539565083307737527272360728012626590607}{12959236999397683492782133631770074387441052} a^{9} + \frac{152893501237876333565788686977467569846447}{6479618499698841746391066815885037193720526} a^{8} - \frac{33822829866540032869792400685134621450587}{1178112454490698499343830330160915853403732} a^{7} - \frac{900457617130024856679073210202692085256825}{3239809249849420873195533407942518596860263} a^{6} + \frac{916968063505449425316844650431073862408411}{3239809249849420873195533407942518596860263} a^{5} - \frac{810045027878723075788745071394718929645538}{3239809249849420873195533407942518596860263} a^{4} - \frac{1749488667411688196679592060347944581621647}{12959236999397683492782133631770074387441052} a^{3} + \frac{701270132540886346584334707851966708375645}{6479618499698841746391066815885037193720526} a^{2} - \frac{669568248369959764730698288256119729023828}{3239809249849420873195533407942518596860263} a + \frac{104666316871948902448562519607146553010185}{294528113622674624835957582540228963350933}$

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:  \( 6091540254.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed