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Properties

Label 20.4.65861868411...0144.5
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 32, -8, -298, 661, -995, 576, 171, -909, 1000, -550, 119, 138, -48, 14, 47, -5, 5, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 + 5*x^17 - 5*x^16 + 47*x^15 + 14*x^14 - 48*x^13 + 138*x^12 + 119*x^11 - 550*x^10 + 1000*x^9 - 909*x^8 + 171*x^7 + 576*x^6 - 995*x^5 + 661*x^4 - 298*x^3 - 8*x^2 + 32*x - 23)
 
gp: K = bnfinit(x^20 - x^19 + 6*x^18 + 5*x^17 - 5*x^16 + 47*x^15 + 14*x^14 - 48*x^13 + 138*x^12 + 119*x^11 - 550*x^10 + 1000*x^9 - 909*x^8 + 171*x^7 + 576*x^6 - 995*x^5 + 661*x^4 - 298*x^3 - 8*x^2 + 32*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 6 x^{18} + 5 x^{17} - 5 x^{16} + 47 x^{15} + 14 x^{14} - 48 x^{13} + 138 x^{12} + 119 x^{11} - 550 x^{10} + 1000 x^{9} - 909 x^{8} + 171 x^{7} + 576 x^{6} - 995 x^{5} + 661 x^{4} - 298 x^{3} - 8 x^{2} + 32 x - 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{8769828279190902708773561} a^{19} - \frac{2059784068430367580858144}{8769828279190902708773561} a^{18} - \frac{44476329188452480705852}{8769828279190902708773561} a^{17} - \frac{1321056147626655026671467}{8769828279190902708773561} a^{16} - \frac{2756439611119501048700892}{8769828279190902708773561} a^{15} - \frac{928837720042617933035592}{8769828279190902708773561} a^{14} + \frac{4044646703735822851839055}{8769828279190902708773561} a^{13} + \frac{3950017602599555018129301}{8769828279190902708773561} a^{12} + \frac{4094232524603528363290365}{8769828279190902708773561} a^{11} - \frac{3982581428998082934485529}{8769828279190902708773561} a^{10} + \frac{977725477801554701680014}{8769828279190902708773561} a^{9} + \frac{2831557902095949678}{65167814340104647357} a^{8} + \frac{1898081564707136325293408}{8769828279190902708773561} a^{7} + \frac{688107742538459451976422}{8769828279190902708773561} a^{6} + \frac{4013235582683583456526513}{8769828279190902708773561} a^{5} - \frac{2325749055074273820995311}{8769828279190902708773561} a^{4} + \frac{2937946754169742647782076}{8769828279190902708773561} a^{3} + \frac{3829526840439633676917347}{8769828279190902708773561} a^{2} - \frac{1082569521570558360516233}{8769828279190902708773561} a + \frac{24609538361193725922000}{381296881703952291685807}$

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:  $11$
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:  \( 259601.408434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
241Data not computed