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Properties

Label 11.3.94640565245...9936.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{8}\cdot 41^{8}\cdot 47^{4}\cdot 3121^{4}$
Root discriminant $1864.46$
Ramified primes $2, 41, 47, 3121$
Class number $4$ (GRH)
Class group $[4]$ (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![-5256, -21908, -40910, -42091, -23660, -5042, 1580, 999, 52, -49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 - 49*x^9 + 52*x^8 + 999*x^7 + 1580*x^6 - 5042*x^5 - 23660*x^4 - 42091*x^3 - 40910*x^2 - 21908*x - 5256)
 
gp: K = bnfinit(x^11 - 4*x^10 - 49*x^9 + 52*x^8 + 999*x^7 + 1580*x^6 - 5042*x^5 - 23660*x^4 - 42091*x^3 - 40910*x^2 - 21908*x - 5256, 1)
 

Normalized defining polynomial

\( x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256 \)

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:  \(946405652456836696490269155592519936=2^{8}\cdot 41^{8}\cdot 47^{4}\cdot 3121^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1864.46$
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:  $2, 41, 47, 3121$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 73468338537500 \) (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: 11.3.946405652456836696490269155592519936.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
41Data not computed
47Data not computed
3121Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.2e4_41e4_47e2_3121e2.55.1c1$5$ $ 2^{4} \cdot 41^{4} \cdot 47^{2} \cdot 3121^{2}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $0$ $1$
5.2e4_41e4_47e2_3121e2.55.1c2$5$ $ 2^{4} \cdot 41^{4} \cdot 47^{2} \cdot 3121^{2}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $0$ $1$
* 10.2e8_41e8_47e4_3121e4.11t5.1c1$10$ $ 2^{8} \cdot 41^{8} \cdot 47^{4} \cdot 3121^{4}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $1$ $2$
10.2e12_41e8_47e6_3121e6.55.1c1$10$ $ 2^{12} \cdot 41^{8} \cdot 47^{6} \cdot 3121^{6}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $1$ $-2$
11.2e12_41e10_47e6_3121e6.12t179.1c1$11$ $ 2^{12} \cdot 41^{10} \cdot 47^{6} \cdot 3121^{6}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $1$ $-1$
12.2e12_41e10_47e6_3121e6.55.1c1$12$ $ 2^{12} \cdot 41^{10} \cdot 47^{6} \cdot 3121^{6}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $1$ $0$
12.2e12_41e10_47e6_3121e6.55.1c2$12$ $ 2^{12} \cdot 41^{10} \cdot 47^{6} \cdot 3121^{6}$ $x^{11} - 4 x^{10} - 49 x^{9} + 52 x^{8} + 999 x^{7} + 1580 x^{6} - 5042 x^{5} - 23660 x^{4} - 42091 x^{3} - 40910 x^{2} - 21908 x - 5256$ $\PSL(2,11)$ (as 11T5) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.