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Properties

Label 20.12.1348323859...4976.5
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $25.50$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, 199, -718, 1186, -766, -1051, 3378, -3655, 1070, 2295, -3664, 2093, 176, -974, 590, -121, -44, 37, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 37*x^18 - 44*x^17 - 121*x^16 + 590*x^15 - 974*x^14 + 176*x^13 + 2093*x^12 - 3664*x^11 + 2295*x^10 + 1070*x^9 - 3655*x^8 + 3378*x^7 - 1051*x^6 - 766*x^5 + 1186*x^4 - 718*x^3 + 199*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 37*x^18 - 44*x^17 - 121*x^16 + 590*x^15 - 974*x^14 + 176*x^13 + 2093*x^12 - 3664*x^11 + 2295*x^10 + 1070*x^9 - 3655*x^8 + 3378*x^7 - 1051*x^6 - 766*x^5 + 1186*x^4 - 718*x^3 + 199*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 37 x^{18} - 44 x^{17} - 121 x^{16} + 590 x^{15} - 974 x^{14} + 176 x^{13} + 2093 x^{12} - 3664 x^{11} + 2295 x^{10} + 1070 x^{9} - 3655 x^{8} + 3378 x^{7} - 1051 x^{6} - 766 x^{5} + 1186 x^{4} - 718 x^{3} + 199 x^{2} - 24 x + 1 \)

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:  \(13483238599952325260241534976=2^{20}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.50$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1110843742630236637616903} a^{19} - \frac{24626496302450054636798}{1110843742630236637616903} a^{18} + \frac{554177470266410445859444}{1110843742630236637616903} a^{17} + \frac{140709554012513371678612}{1110843742630236637616903} a^{16} - \frac{321203051881413419909928}{1110843742630236637616903} a^{15} - \frac{350488336293108702324110}{1110843742630236637616903} a^{14} + \frac{6717172213131370164000}{1110843742630236637616903} a^{13} + \frac{256728119693396593705653}{1110843742630236637616903} a^{12} + \frac{209285876213402195004096}{1110843742630236637616903} a^{11} - \frac{61725982018281030953930}{1110843742630236637616903} a^{10} + \frac{489826918516075378281154}{1110843742630236637616903} a^{9} - \frac{545347995286401056111072}{1110843742630236637616903} a^{8} - \frac{45702687795433828619421}{1110843742630236637616903} a^{7} - \frac{44565483395350588285614}{1110843742630236637616903} a^{6} - \frac{372880706470053849494636}{1110843742630236637616903} a^{5} - \frac{226931294258067979512962}{1110843742630236637616903} a^{4} + \frac{549497691867231252065923}{1110843742630236637616903} a^{3} - \frac{64113403316311185944257}{1110843742630236637616903} a^{2} - \frac{341781715327849070650342}{1110843742630236637616903} a + \frac{305761224760909019018616}{1110843742630236637616903}$

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:  \( 6150965.25836 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

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 t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ 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 ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
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}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$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}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$