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Properties

Label 20.4.36487627390...3376.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{3}$
Root discriminant $21.29$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -189, -364, 164, 1857, 2072, -435, -2141, -1408, -184, 418, 370, 91, 51, 84, 47, 16, -2, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - 2*x^17 + 16*x^16 + 47*x^15 + 84*x^14 + 51*x^13 + 91*x^12 + 370*x^11 + 418*x^10 - 184*x^9 - 1408*x^8 - 2141*x^7 - 435*x^6 + 2072*x^5 + 1857*x^4 + 164*x^3 - 364*x^2 - 189*x - 23)
 
gp: K = bnfinit(x^20 - 6*x^18 - 2*x^17 + 16*x^16 + 47*x^15 + 84*x^14 + 51*x^13 + 91*x^12 + 370*x^11 + 418*x^10 - 184*x^9 - 1408*x^8 - 2141*x^7 - 435*x^6 + 2072*x^5 + 1857*x^4 + 164*x^3 - 364*x^2 - 189*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - 2 x^{17} + 16 x^{16} + 47 x^{15} + 84 x^{14} + 51 x^{13} + 91 x^{12} + 370 x^{11} + 418 x^{10} - 184 x^{9} - 1408 x^{8} - 2141 x^{7} - 435 x^{6} + 2072 x^{5} + 1857 x^{4} + 164 x^{3} - 364 x^{2} - 189 x - 23 \)

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:  \(364876273903737477578853376=2^{10}\cdot 11^{17}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.29$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{14} - \frac{3}{11} a^{13} + \frac{1}{11} a^{12} - \frac{1}{11} a^{10} + \frac{3}{11} a^{9} + \frac{2}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} + \frac{1}{11} a^{5} - \frac{3}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{14} + \frac{4}{11} a^{12} - \frac{1}{11} a^{11} - \frac{1}{11} a^{10} + \frac{3}{11} a^{9} + \frac{2}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{6} + \frac{4}{11} a^{5} - \frac{3}{11} a^{4} - \frac{4}{11} a^{3} - \frac{5}{11} a^{2} + \frac{5}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} + \frac{1}{11} a^{14} + \frac{2}{11} a^{13} - \frac{4}{11} a^{12} - \frac{1}{11} a^{11} - \frac{5}{11} a^{10} + \frac{4}{11} a^{9} - \frac{1}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{2}{11} a^{12} - \frac{5}{11} a^{11} + \frac{5}{11} a^{10} - \frac{4}{11} a^{9} + \frac{1}{11} a^{8} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{4} - \frac{5}{11} a^{3} - \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{63208632715936398434138263} a^{19} - \frac{1508238185338115904740561}{63208632715936398434138263} a^{18} - \frac{2326270438264240530537435}{63208632715936398434138263} a^{17} + \frac{1236599712407574081059936}{63208632715936398434138263} a^{16} + \frac{2688909852466901433649649}{63208632715936398434138263} a^{15} + \frac{25012516069919993333597929}{63208632715936398434138263} a^{14} - \frac{27092716169656363116995448}{63208632715936398434138263} a^{13} + \frac{1791649421231906115521156}{5746239337812399857648933} a^{12} + \frac{15643689499389550729385058}{63208632715936398434138263} a^{11} - \frac{19048822968875052291839400}{63208632715936398434138263} a^{10} + \frac{14390409219650705110255005}{63208632715936398434138263} a^{9} + \frac{22767918467674741966550211}{63208632715936398434138263} a^{8} - \frac{5390583914688030499878959}{63208632715936398434138263} a^{7} + \frac{18097654510691001605376447}{63208632715936398434138263} a^{6} - \frac{14776418754513876834836580}{63208632715936398434138263} a^{5} - \frac{15470870417683400187624865}{63208632715936398434138263} a^{4} + \frac{13933096826049829213216582}{63208632715936398434138263} a^{3} + \frac{2189200483429354137565701}{63208632715936398434138263} a^{2} + \frac{1408077027785622447660383}{63208632715936398434138263} a - \frac{24624182377900727582388628}{63208632715936398434138263}$

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:  \( 256871.832795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed