↑

Properties

Label 20.8.27296637038...4832.1
Degree $20$
Signature $[8, 6]$
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![-4, -312, -808, -5848, 10733, 3642, -12985, 11226, -14282, 21264, -20613, 14196, -6799, 2142, -481, 24, 57, -42, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 24*x^18 - 42*x^17 + 57*x^16 + 24*x^15 - 481*x^14 + 2142*x^13 - 6799*x^12 + 14196*x^11 - 20613*x^10 + 21264*x^9 - 14282*x^8 + 11226*x^7 - 12985*x^6 + 3642*x^5 + 10733*x^4 - 5848*x^3 - 808*x^2 - 312*x - 4)
 
gp: K = bnfinit(x^20 - 8*x^19 + 24*x^18 - 42*x^17 + 57*x^16 + 24*x^15 - 481*x^14 + 2142*x^13 - 6799*x^12 + 14196*x^11 - 20613*x^10 + 21264*x^9 - 14282*x^8 + 11226*x^7 - 12985*x^6 + 3642*x^5 + 10733*x^4 - 5848*x^3 - 808*x^2 - 312*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 24 x^{18} - 42 x^{17} + 57 x^{16} + 24 x^{15} - 481 x^{14} + 2142 x^{13} - 6799 x^{12} + 14196 x^{11} - 20613 x^{10} + 21264 x^{9} - 14282 x^{8} + 11226 x^{7} - 12985 x^{6} + 3642 x^{5} + 10733 x^{4} - 5848 x^{3} - 808 x^{2} - 312 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{16} + \frac{1}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{5} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{61922} a^{18} + \frac{50}{30961} a^{17} - \frac{1191}{30961} a^{16} - \frac{2132}{4423} a^{15} - \frac{3267}{61922} a^{14} - \frac{2022}{30961} a^{13} - \frac{2773}{8846} a^{12} - \frac{11260}{30961} a^{11} + \frac{22507}{61922} a^{10} + \frac{8874}{30961} a^{9} - \frac{25253}{61922} a^{8} + \frac{6595}{30961} a^{7} + \frac{13590}{30961} a^{6} - \frac{13072}{30961} a^{5} - \frac{3899}{8846} a^{4} + \frac{1}{7} a^{3} + \frac{20667}{61922} a^{2} + \frac{1681}{4423} a + \frac{12918}{30961}$, $\frac{1}{20784839358524920959370206042610628} a^{19} + \frac{1101012083664399793706376270}{742315691375890034263221644378951} a^{18} + \frac{704871520864291024856470464843427}{10392419679262460479685103021305314} a^{17} + \frac{156107848781025943038471656014209}{1484631382751780068526443288757902} a^{16} + \frac{2493046507420027096468904227704621}{20784839358524920959370206042610628} a^{15} + \frac{1332887178930616855889668316933877}{5196209839631230239842551510652657} a^{14} + \frac{3666526435378289768212669331849209}{20784839358524920959370206042610628} a^{13} - \frac{2703559185771677746874484930119017}{10392419679262460479685103021305314} a^{12} + \frac{5452308087149256491054737789293627}{20784839358524920959370206042610628} a^{11} - \frac{800244299206578835985151990749872}{5196209839631230239842551510652657} a^{10} + \frac{4849191441377256337962080721459329}{20784839358524920959370206042610628} a^{9} + \frac{1353637856131845464132839300341210}{5196209839631230239842551510652657} a^{8} - \frac{98762036084123237001043311438029}{742315691375890034263221644378951} a^{7} - \frac{614822220250084048107954410840739}{10392419679262460479685103021305314} a^{6} + \frac{8530238560095928322998599602080535}{20784839358524920959370206042610628} a^{5} + \frac{959938200277953561631938031339929}{10392419679262460479685103021305314} a^{4} + \frac{721529129255703780288734984132325}{2969262765503560137052886577515804} a^{3} - \frac{1144206232886000397612244608898366}{5196209839631230239842551510652657} a^{2} + \frac{630601065890654308546403485453845}{1484631382751780068526443288757902} a - \frac{2191965087545249419525022862584316}{5196209839631230239842551510652657}$

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:  $13$
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:  \( 3878715721.05 \) (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 ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13Data not computed