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Properties

Label 20.8.65861868411...0144.4
Degree $20$
Signature $[8, 6]$
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![1, -17, 107, -315, 373, 23, -81, -630, 327, 520, 198, -342, -598, 489, 175, -232, 18, 35, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 5*x^18 + 35*x^17 + 18*x^16 - 232*x^15 + 175*x^14 + 489*x^13 - 598*x^12 - 342*x^11 + 198*x^10 + 520*x^9 + 327*x^8 - 630*x^7 - 81*x^6 + 23*x^5 + 373*x^4 - 315*x^3 + 107*x^2 - 17*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 5*x^18 + 35*x^17 + 18*x^16 - 232*x^15 + 175*x^14 + 489*x^13 - 598*x^12 - 342*x^11 + 198*x^10 + 520*x^9 + 327*x^8 - 630*x^7 - 81*x^6 + 23*x^5 + 373*x^4 - 315*x^3 + 107*x^2 - 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 5 x^{18} + 35 x^{17} + 18 x^{16} - 232 x^{15} + 175 x^{14} + 489 x^{13} - 598 x^{12} - 342 x^{11} + 198 x^{10} + 520 x^{9} + 327 x^{8} - 630 x^{7} - 81 x^{6} + 23 x^{5} + 373 x^{4} - 315 x^{3} + 107 x^{2} - 17 x + 1 \)

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:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5365893474988143878771} a^{19} - \frac{2077369466970046727917}{5365893474988143878771} a^{18} + \frac{2660388039547402017242}{5365893474988143878771} a^{17} + \frac{1210216018863604841622}{5365893474988143878771} a^{16} - \frac{82437442107097231828}{5365893474988143878771} a^{15} + \frac{1665447322173314797963}{5365893474988143878771} a^{14} - \frac{843870408365769409870}{5365893474988143878771} a^{13} + \frac{1954244837267379670528}{5365893474988143878771} a^{12} + \frac{786664022976386969076}{5365893474988143878771} a^{11} + \frac{2201267752224888658302}{5365893474988143878771} a^{10} - \frac{2305800696550401145262}{5365893474988143878771} a^{9} - \frac{1502845413200624078749}{5365893474988143878771} a^{8} - \frac{857650385023257187774}{5365893474988143878771} a^{7} - \frac{2608792173831008750918}{5365893474988143878771} a^{6} - \frac{1704704708725223837491}{5365893474988143878771} a^{5} + \frac{254503951786043625623}{5365893474988143878771} a^{4} + \frac{123129589296558347924}{5365893474988143878771} a^{3} - \frac{711913217022270845138}{5365893474988143878771} a^{2} - \frac{2257877070492426332892}{5365893474988143878771} a - \frac{841085019286664590378}{5365893474988143878771}$

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:  \( 576329.020165 \) (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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
241Data not computed