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Properties

Label 11.3.17019463777...3696.3
Degree $11$
Signature $[3, 4]$
Discriminant $2^{6}\cdot 3^{14}\cdot 11^{18}$
Root discriminant $298.91$
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![19955895, 8323854, -16889246, -8647353, -1063260, 111650, 49698, 6204, -319, -132, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 132*x^9 - 319*x^8 + 6204*x^7 + 49698*x^6 + 111650*x^5 - 1063260*x^4 - 8647353*x^3 - 16889246*x^2 + 8323854*x + 19955895)
 
gp: K = bnfinit(x^11 - 132*x^9 - 319*x^8 + 6204*x^7 + 49698*x^6 + 111650*x^5 - 1063260*x^4 - 8647353*x^3 - 16889246*x^2 + 8323854*x + 19955895, 1)
 

Normalized defining polynomial

\( x^{11} - 132 x^{9} - 319 x^{8} + 6204 x^{7} + 49698 x^{6} + 111650 x^{5} - 1063260 x^{4} - 8647353 x^{3} - 16889246 x^{2} + 8323854 x + 19955895 \)

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:  \(1701946377791783994524613696=2^{6}\cdot 3^{14}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $298.91$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{22} a^{6} - \frac{2}{11} a^{5} + \frac{3}{22} a^{4} + \frac{1}{22} a^{3} - \frac{3}{11} a^{2} - \frac{5}{22} a - \frac{4}{11}$, $\frac{1}{44} a^{7} - \frac{1}{22} a^{5} - \frac{9}{44} a^{4} + \frac{9}{44} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{1}{44}$, $\frac{1}{44} a^{8} + \frac{5}{44} a^{5} + \frac{15}{44} a^{4} - \frac{4}{11} a^{3} - \frac{9}{22} a^{2} - \frac{9}{44} a + \frac{3}{22}$, $\frac{1}{44} a^{9} - \frac{1}{44} a^{6} - \frac{5}{44} a^{5} + \frac{5}{22} a^{4} + \frac{5}{11} a^{3} - \frac{17}{44} a^{2} - \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{901490848590168550476136} a^{10} - \frac{5714087829372096359917}{901490848590168550476136} a^{9} - \frac{7400313301873093255369}{901490848590168550476136} a^{8} + \frac{1086289872174806828961}{225372712147542137619034} a^{7} - \frac{914859613875209583903}{225372712147542137619034} a^{6} + \frac{52299122324917442120351}{225372712147542137619034} a^{5} - \frac{169656878117632156984305}{450745424295084275238068} a^{4} + \frac{14646440377374959027250}{112686356073771068809517} a^{3} - \frac{27207559110085363796907}{81953713508197140952376} a^{2} + \frac{123530214515280066979357}{901490848590168550476136} a + \frac{358827759568664552375087}{901490848590168550476136}$

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:  \( 78303121703.0 \) (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: 11.3.1701946377791783994524613696.4

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
$11$11.11.18.3$x^{11} + 22 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$