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Properties

Label 11.3.19364367676...3856.6
Degree $11$
Signature $[3, 4]$
Discriminant $2^{16}\cdot 3^{12}\cdot 11^{18}$
Root discriminant $459.68$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{11}$ (as 11T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![214716744, 15267780, -3348312, -37752, -932910, -21659, 15972, 4719, -550, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 33*x^9 - 550*x^8 + 4719*x^7 + 15972*x^6 - 21659*x^5 - 932910*x^4 - 37752*x^3 - 3348312*x^2 + 15267780*x + 214716744)
 
gp: K = bnfinit(x^11 - 33*x^9 - 550*x^8 + 4719*x^7 + 15972*x^6 - 21659*x^5 - 932910*x^4 - 37752*x^3 - 3348312*x^2 + 15267780*x + 214716744, 1)
 

Normalized defining polynomial

\( x^{11} - 33 x^{9} - 550 x^{8} + 4719 x^{7} + 15972 x^{6} - 21659 x^{5} - 932910 x^{4} - 37752 x^{3} - 3348312 x^{2} + 15267780 x + 214716744 \)

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:  \(193643676762087423377022713856=2^{16}\cdot 3^{12}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $459.68$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{5} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{3072} a^{9} + \frac{7}{512} a^{8} - \frac{9}{512} a^{7} + \frac{41}{768} a^{6} + \frac{31}{1024} a^{5} + \frac{27}{512} a^{4} - \frac{191}{768} a^{3} - \frac{1}{128} a^{2} + \frac{79}{256} a + \frac{59}{128}$, $\frac{1}{114439605638547492347461632} a^{10} + \frac{4755022212230845507303}{38146535212849164115820544} a^{9} + \frac{51493151725147603233305}{4768316901606145514477568} a^{8} - \frac{1006692710575268563249271}{57219802819273746173730816} a^{7} - \frac{122206044938342386515423}{12715511737616388038606848} a^{6} - \frac{648206328762143873932597}{38146535212849164115820544} a^{5} + \frac{7158221475639404983983677}{57219802819273746173730816} a^{4} - \frac{385377158528394281109097}{9536633803212291028955136} a^{3} + \frac{3778634169171687277551865}{9536633803212291028955136} a^{2} + \frac{1419203455926302691391771}{9536633803212291028955136} a - \frac{689194863679378049548989}{1589438967202048504825856}$

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:  \( 3728832834870 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{11}$ (as 11T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19958400
The 31 conjugacy class representatives for $A_{11}$
Character table for $A_{11}$ is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.8.6$x^{4} + 6 x^{2} + 4 x + 2$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.9.12$x^{6} + 3 x^{4} + 3$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
$11$11.11.18.2$x^{11} + 77 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$