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Properties

Label 11.3.30113614963...3296.2
Degree $11$
Signature $[3, 4]$
Discriminant $2^{16}\cdot 11^{16}$
Root discriminant $89.66$
Ramified primes $2, 11$
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![-60672, -352000, 16192, 26048, 4048, -1408, -132, 396, -22, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 33*x^9 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672)
 
gp: K = bnfinit(x^11 - 33*x^9 - 22*x^8 + 396*x^7 - 132*x^6 - 1408*x^5 + 4048*x^4 + 26048*x^3 + 16192*x^2 - 352000*x - 60672, 1)
 

Normalized defining polynomial

\( x^{11} - 33 x^{9} - 22 x^{8} + 396 x^{7} - 132 x^{6} - 1408 x^{5} + 4048 x^{4} + 26048 x^{3} + 16192 x^{2} - 352000 x - 60672 \)

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:  \(3011361496339065143296=2^{16}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.66$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{6} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{7} + \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{300312335380284544} a^{10} - \frac{184998999493835}{37539041922535568} a^{9} - \frac{2515890646991445}{300312335380284544} a^{8} + \frac{246793817321013}{150156167690142272} a^{7} + \frac{14631225182049}{2346190120158473} a^{6} + \frac{2186754944976307}{75078083845071136} a^{5} + \frac{491712753829421}{18769520961267784} a^{4} - \frac{209880365091193}{4692380240316946} a^{3} + \frac{564746769856721}{9384760480633892} a^{2} + \frac{1428067545074557}{4692380240316946} a - \frac{1062364761543537}{2346190120158473}$

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:  \( 52170956.4414 \) (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: 11.3.3011361496339065143296.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$11$11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$