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Properties

Label 20.8.26099879633...0256.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{2}$
Root discriminant $26.35$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -112, -20, 642, 639, -1306, -1683, 892, 1249, -10, 132, 86, 43, 90, -71, -12, -5, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 4*x^17 - 5*x^16 - 12*x^15 - 71*x^14 + 90*x^13 + 43*x^12 + 86*x^11 + 132*x^10 - 10*x^9 + 1249*x^8 + 892*x^7 - 1683*x^6 - 1306*x^5 + 639*x^4 + 642*x^3 - 20*x^2 - 112*x - 23)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 4*x^17 - 5*x^16 - 12*x^15 - 71*x^14 + 90*x^13 + 43*x^12 + 86*x^11 + 132*x^10 - 10*x^9 + 1249*x^8 + 892*x^7 - 1683*x^6 - 1306*x^5 + 639*x^4 + 642*x^3 - 20*x^2 - 112*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 4 x^{17} - 5 x^{16} - 12 x^{15} - 71 x^{14} + 90 x^{13} + 43 x^{12} + 86 x^{11} + 132 x^{10} - 10 x^{9} + 1249 x^{8} + 892 x^{7} - 1683 x^{6} - 1306 x^{5} + 639 x^{4} + 642 x^{3} - 20 x^{2} - 112 x - 23 \)

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:  \(26099879633934179709805920256=2^{30}\cdot 11^{16}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.35$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{3}{23} a^{17} + \frac{2}{23} a^{16} + \frac{6}{23} a^{15} + \frac{4}{23} a^{14} + \frac{6}{23} a^{13} - \frac{1}{23} a^{12} - \frac{2}{23} a^{11} + \frac{3}{23} a^{10} - \frac{1}{23} a^{9} + \frac{6}{23} a^{8} + \frac{11}{23} a^{7} - \frac{5}{23} a^{6} + \frac{2}{23} a^{5} - \frac{9}{23} a^{4} + \frac{6}{23} a^{3} + \frac{2}{23} a^{2} - \frac{5}{23} a$, $\frac{1}{35084260470153174097493609} a^{19} + \frac{649800934322335445527887}{35084260470153174097493609} a^{18} - \frac{10012184132012117962245199}{35084260470153174097493609} a^{17} - \frac{1394014538531888344735736}{35084260470153174097493609} a^{16} - \frac{12582504019438930159895436}{35084260470153174097493609} a^{15} - \frac{11484053604628289724869618}{35084260470153174097493609} a^{14} - \frac{4448060718593501283975744}{35084260470153174097493609} a^{13} + \frac{11860229532940345137117123}{35084260470153174097493609} a^{12} + \frac{8065855082339580488250204}{35084260470153174097493609} a^{11} - \frac{1804607767420951885218208}{35084260470153174097493609} a^{10} - \frac{13048348107799211487104001}{35084260470153174097493609} a^{9} - \frac{8039954383161360184728636}{35084260470153174097493609} a^{8} + \frac{5764964327791703755349715}{35084260470153174097493609} a^{7} + \frac{43274160199061022027831}{35084260470153174097493609} a^{6} - \frac{15504996594853711393901888}{35084260470153174097493609} a^{5} - \frac{13449680880198665452995641}{35084260470153174097493609} a^{4} - \frac{12744407722609323083878943}{35084260470153174097493609} a^{3} - \frac{14381211908275479357547029}{35084260470153174097493609} a^{2} + \frac{16797128444848403822224213}{35084260470153174097493609} a - \frac{340020227328115720401265}{1525402629137094525977983}$

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:  \( 4918338.47388 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$