↑

Properties

Label 20.0.15583578925...2688.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{5}\cdot 11^{19}$
Root discriminant $25.68$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group 20T130

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2673, 0, 9801, 0, 14256, 0, 9801, 0, 2442, 0, -737, 0, -572, 0, -66, 0, 33, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 11*x^18 + 33*x^16 - 66*x^14 - 572*x^12 - 737*x^10 + 2442*x^8 + 9801*x^6 + 14256*x^4 + 9801*x^2 + 2673)
 
gp: K = bnfinit(x^20 + 11*x^18 + 33*x^16 - 66*x^14 - 572*x^12 - 737*x^10 + 2442*x^8 + 9801*x^6 + 14256*x^4 + 9801*x^2 + 2673, 1)
 

Normalized defining polynomial

\( x^{20} + 11 x^{18} + 33 x^{16} - 66 x^{14} - 572 x^{12} - 737 x^{10} + 2442 x^{8} + 9801 x^{6} + 14256 x^{4} + 9801 x^{2} + 2673 \)

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:  \(15583578925526925703866482688=2^{20}\cdot 3^{5}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{3} a^{8} + \frac{4}{9} a^{6} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{3} a^{9} + \frac{4}{9} a^{7} - \frac{2}{9} a^{5}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} + \frac{2}{9} a^{10} + \frac{4}{27} a^{8} - \frac{2}{27} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{15} + \frac{2}{9} a^{11} + \frac{4}{27} a^{9} - \frac{2}{27} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{394123239} a^{18} + \frac{6059282}{394123239} a^{16} + \frac{269936}{5711931} a^{14} + \frac{2917574}{131374413} a^{12} + \frac{52303468}{394123239} a^{10} + \frac{84399367}{394123239} a^{8} - \frac{58620589}{131374413} a^{6} + \frac{5498684}{43791471} a^{4} + \frac{1882816}{14597157} a^{2} - \frac{1082931}{4865719}$, $\frac{1}{394123239} a^{19} + \frac{6059282}{394123239} a^{17} + \frac{269936}{5711931} a^{15} + \frac{2917574}{131374413} a^{13} + \frac{52303468}{394123239} a^{11} + \frac{84399367}{394123239} a^{9} - \frac{58620589}{131374413} a^{7} + \frac{5498684}{43791471} a^{5} + \frac{1882816}{14597157} a^{3} - \frac{1082931}{4865719} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{114128}{43791471} a^{18} + \frac{4598258}{131374413} a^{16} + \frac{767177}{5711931} a^{14} - \frac{6320003}{43791471} a^{12} - \frac{10612919}{4865719} a^{10} - \frac{463612882}{131374413} a^{8} + \frac{1216886951}{131374413} a^{6} + \frac{1726088864}{43791471} a^{4} + \frac{715075547}{14597157} a^{2} + \frac{99995301}{4865719} \) (order $22$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3282524.64052 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\)

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed