↑

Properties

Label 20.8.19788394110...6549.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 23\cdot 263^{2}\cdot 2707$
Root discriminant $20.65$
Ramified primes $11, 23, 263, 2707$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 58, 71, -192, -331, 201, 738, -3, -1067, -98, 1060, -51, -680, 182, 240, -132, -21, 34, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 132*x^16 + 240*x^15 + 182*x^14 - 680*x^13 - 51*x^12 + 1060*x^11 - 98*x^10 - 1067*x^9 - 3*x^8 + 738*x^7 + 201*x^6 - 331*x^5 - 192*x^4 + 71*x^3 + 58*x^2 - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 132*x^16 + 240*x^15 + 182*x^14 - 680*x^13 - 51*x^12 + 1060*x^11 - 98*x^10 - 1067*x^9 - 3*x^8 + 738*x^7 + 201*x^6 - 331*x^5 - 192*x^4 + 71*x^3 + 58*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 34 x^{18} - 21 x^{17} - 132 x^{16} + 240 x^{15} + 182 x^{14} - 680 x^{13} - 51 x^{12} + 1060 x^{11} - 98 x^{10} - 1067 x^{9} - 3 x^{8} + 738 x^{7} + 201 x^{6} - 331 x^{5} - 192 x^{4} + 71 x^{3} + 58 x^{2} - 1 \)

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:  \(197883941107619837212856549=11^{16}\cdot 23\cdot 263^{2}\cdot 2707\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.65$
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:  $11, 23, 263, 2707$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{14} - \frac{1}{11} a^{13} + \frac{2}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{5} - \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{2}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{5}{11} a^{14} - \frac{3}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{4}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} + \frac{3}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{17} + \frac{1}{11} a^{14} + \frac{3}{11} a^{13} - \frac{4}{11} a^{12} + \frac{5}{11} a^{11} - \frac{1}{11} a^{10} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} + \frac{4}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{18} + \frac{5}{11} a^{14} - \frac{3}{11} a^{13} + \frac{3}{11} a^{12} - \frac{5}{11} a^{11} + \frac{3}{11} a^{10} + \frac{2}{11} a^{9} - \frac{1}{11} a^{8} - \frac{3}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{4} + \frac{5}{11} a^{3} + \frac{5}{11} a^{2} + \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{19} - \frac{4}{11} a^{14} - \frac{3}{11} a^{13} - \frac{4}{11} a^{12} + \frac{5}{11} a^{11} + \frac{2}{11} a^{10} - \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{2}{11} a^{7} + \frac{5}{11} a^{5} + \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{5}{11}$

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:  \( 302464.100794 \) (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

\(\Q(\zeta_{11})^+\), 10.4.56376385703.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
263Data not computed
2707Data not computed