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Properties

Label 11.3.24392028120...6976.6
Degree $11$
Signature $[3, 4]$
Discriminant $2^{16}\cdot 3^{4}\cdot 11^{16}$
Root discriminant $133.69$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86888, 20592, 31900, -15554, 3432, -7150, -2024, -264, -154, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 - 154*x^8 - 264*x^7 - 2024*x^6 - 7150*x^5 + 3432*x^4 - 15554*x^3 + 31900*x^2 + 20592*x + 86888)
 
gp: K = bnfinit(x^11 - 55*x^9 - 154*x^8 - 264*x^7 - 2024*x^6 - 7150*x^5 + 3432*x^4 - 15554*x^3 + 31900*x^2 + 20592*x + 86888, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} - 154 x^{8} - 264 x^{7} - 2024 x^{6} - 7150 x^{5} + 3432 x^{4} - 15554 x^{3} + 31900 x^{2} + 20592 x + 86888 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(243920281203464276606976=2^{16}\cdot 3^{4}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.69$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{297} a^{6} + \frac{16}{297} a^{5} + \frac{26}{297} a^{4} - \frac{20}{297} a^{3} + \frac{59}{297} a^{2} - \frac{50}{297} a - \frac{32}{297}$, $\frac{1}{594} a^{7} - \frac{1}{594} a^{6} + \frac{1}{33} a^{5} + \frac{1}{9} a^{4} - \frac{5}{99} a^{3} - \frac{38}{99} a^{2} + \frac{46}{297} a + \frac{41}{297}$, $\frac{1}{594} a^{8} - \frac{1}{594} a^{6} - \frac{1}{99} a^{5} - \frac{2}{33} a^{4} - \frac{16}{99} a^{3} + \frac{94}{297} a^{2} - \frac{19}{99} a + \frac{32}{297}$, $\frac{1}{5346} a^{9} - \frac{1}{1782} a^{8} - \frac{1}{1782} a^{7} + \frac{5}{5346} a^{6} - \frac{26}{891} a^{5} - \frac{20}{297} a^{4} + \frac{10}{2673} a^{3} + \frac{26}{81} a^{2} - \frac{409}{891} a - \frac{274}{2673}$, $\frac{1}{27593731836} a^{10} + \frac{19823}{1254260538} a^{9} + \frac{8177}{1021990068} a^{8} - \frac{201436}{6898432959} a^{7} + \frac{202663}{1254260538} a^{6} - \frac{10968707}{2299477653} a^{5} - \frac{332797}{39084606} a^{4} + \frac{614456662}{6898432959} a^{3} + \frac{665697991}{4598955306} a^{2} - \frac{384406151}{6898432959} a + \frac{2293962866}{6898432959}$

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:  $6$
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:  \( 2824285089.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,11)$ (as 11T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed
Arithmetically equvalently sibling: 11.3.243920281203464276606976.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.9$x^{6} + 6 x^{4} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$