↑

Properties

Label 11.3.29406788376...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{18}\cdot 3^{8}\cdot 5^{6}\cdot 149^{6}$
Root discriminant $254.81$
Ramified primes $2, 3, 5, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{11}$ (as 11T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-708588, 0, -406782, 330237, -5832, -83592, -2592, 4806, 144, -114, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 114*x^9 + 144*x^8 + 4806*x^7 - 2592*x^6 - 83592*x^5 - 5832*x^4 + 330237*x^3 - 406782*x^2 - 708588)
 
gp: K = bnfinit(x^11 - 2*x^10 - 114*x^9 + 144*x^8 + 4806*x^7 - 2592*x^6 - 83592*x^5 - 5832*x^4 + 330237*x^3 - 406782*x^2 - 708588, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - 114 x^{9} + 144 x^{8} + 4806 x^{7} - 2592 x^{6} - 83592 x^{5} - 5832 x^{4} + 330237 x^{3} - 406782 x^{2} - 708588 \)

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:  \(294067883764486803456000000=2^{18}\cdot 3^{8}\cdot 5^{6}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.81$
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:  $2, 3, 5, 149$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{4} + \frac{1}{27} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{5} + \frac{1}{81} a^{4} - \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{243} a^{6} + \frac{1}{243} a^{5} - \frac{1}{81} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{729} a^{7} + \frac{1}{729} a^{6} - \frac{1}{243} a^{5} + \frac{1}{27} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{2187} a^{8} + \frac{1}{2187} a^{7} - \frac{1}{729} a^{6} + \frac{1}{81} a^{4} + \frac{1}{27} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{32805} a^{9} - \frac{2}{32805} a^{8} - \frac{1}{2187} a^{7} - \frac{2}{1215} a^{6} + \frac{1}{405} a^{5} - \frac{1}{135} a^{4} - \frac{1}{135} a^{3} + \frac{4}{45} a^{2} + \frac{2}{5}$, $\frac{1}{984150} a^{10} - \frac{7}{492075} a^{9} - \frac{7}{54675} a^{8} + \frac{37}{54675} a^{7} - \frac{34}{18225} a^{6} - \frac{4}{1215} a^{5} + \frac{28}{2025} a^{4} - \frac{4}{75} a^{3} + \frac{59}{450} a^{2} + \frac{11}{75} a + \frac{6}{25}$

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:  \( 25994623180.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 11T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

Intermediate fields

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

Sibling fields

Degree 12 sibling: data not computed
Degree 22 sibling: 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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.16.74$x^{8} + 14 x^{6} + 20$$8$$1$$16$$\textrm{GL(2,3)}$$[4/3, 4/3, 3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.4.3.4$x^{4} + 1192$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.4$x^{4} + 1192$$4$$1$$3$$C_4$$[\ ]_{4}$