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Properties

Label 20.12.3396666404...5184.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $26.70$
Ramified primes $2, 11, 89$
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, -84, 479, -730, -704, 3768, -4750, 534, 5386, -6912, 3249, 1380, -3155, 2218, -728, -92, 241, -138, 47, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 241*x^16 - 92*x^15 - 728*x^14 + 2218*x^13 - 3155*x^12 + 1380*x^11 + 3249*x^10 - 6912*x^9 + 5386*x^8 + 534*x^7 - 4750*x^6 + 3768*x^5 - 704*x^4 - 730*x^3 + 479*x^2 - 84*x - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 241*x^16 - 92*x^15 - 728*x^14 + 2218*x^13 - 3155*x^12 + 1380*x^11 + 3249*x^10 - 6912*x^9 + 5386*x^8 + 534*x^7 - 4750*x^6 + 3768*x^5 - 704*x^4 - 730*x^3 + 479*x^2 - 84*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 47 x^{18} - 138 x^{17} + 241 x^{16} - 92 x^{15} - 728 x^{14} + 2218 x^{13} - 3155 x^{12} + 1380 x^{11} + 3249 x^{10} - 6912 x^{9} + 5386 x^{8} + 534 x^{7} - 4750 x^{6} + 3768 x^{5} - 704 x^{4} - 730 x^{3} + 479 x^{2} - 84 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33966664043402470640067805184=2^{20}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2190706864193} a^{18} - \frac{9}{2190706864193} a^{17} - \frac{737466690808}{2190706864193} a^{16} - \frac{672387065911}{2190706864193} a^{15} - \frac{736893035804}{2190706864193} a^{14} + \frac{494723421909}{2190706864193} a^{13} + \frac{485804408906}{2190706864193} a^{12} + \frac{588234553019}{2190706864193} a^{11} - \frac{944890694460}{2190706864193} a^{10} + \frac{1065237729613}{2190706864193} a^{9} + \frac{685366501983}{2190706864193} a^{8} - \frac{452122710365}{2190706864193} a^{7} + \frac{889369760657}{2190706864193} a^{6} + \frac{15224772551}{2190706864193} a^{5} + \frac{358561034649}{2190706864193} a^{4} - \frac{327350783150}{2190706864193} a^{3} - \frac{765780797863}{2190706864193} a^{2} + \frac{54369595082}{2190706864193} a - \frac{539207034441}{2190706864193}$, $\frac{1}{50386257876439} a^{19} + \frac{2}{50386257876439} a^{18} - \frac{18263121604451}{50386257876439} a^{17} - \frac{4403106936413}{50386257876439} a^{16} + \frac{122625372180}{2190706864193} a^{15} - \frac{235669265554}{2190706864193} a^{14} + \frac{3737055185712}{50386257876439} a^{13} - \frac{11593571862559}{50386257876439} a^{12} + \frac{20860637438100}{50386257876439} a^{11} - \frac{11519266773640}{50386257876439} a^{10} - \frac{18266914570976}{50386257876439} a^{9} + \frac{1070111466304}{2190706864193} a^{8} + \frac{15632381724379}{50386257876439} a^{7} - \frac{9918069637959}{50386257876439} a^{6} - \frac{19190328245027}{50386257876439} a^{5} - \frac{2955299994590}{50386257876439} a^{4} + \frac{6586894908452}{50386257876439} a^{3} - \frac{3987805453025}{50386257876439} a^{2} + \frac{13203099696619}{50386257876439} a + \frac{2831550077921}{50386257876439}$

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:  $15$
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:  \( 10041308.2795 \) (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

\(\Q(\zeta_{11})^+\), 10.10.19535810978816.1

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 $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$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}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$