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Properties

Label 11.3.54882063270...5696.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{14}\cdot 3^{6}\cdot 11^{16}$
Root discriminant $143.92$
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![32768, 0, -19712, 30976, 4224, -7744, -572, 132, -22, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 + 44*x^9 - 22*x^8 + 132*x^7 - 572*x^6 - 7744*x^5 + 4224*x^4 + 30976*x^3 - 19712*x^2 + 32768)
 
gp: K = bnfinit(x^11 + 44*x^9 - 22*x^8 + 132*x^7 - 572*x^6 - 7744*x^5 + 4224*x^4 + 30976*x^3 - 19712*x^2 + 32768, 1)
 

Normalized defining polynomial

\( x^{11} + 44 x^{9} - 22 x^{8} + 132 x^{7} - 572 x^{6} - 7744 x^{5} + 4224 x^{4} + 30976 x^{3} - 19712 x^{2} + 32768 \)

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:  \(548820632707794622365696=2^{14}\cdot 3^{6}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.92$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{5} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{72} a^{6} - \frac{1}{36} a^{5} - \frac{1}{18} a^{4} - \frac{7}{36} a^{3} + \frac{1}{9} a^{2} - \frac{5}{18} a - \frac{4}{9}$, $\frac{1}{864} a^{7} + \frac{1}{216} a^{5} - \frac{11}{432} a^{4} + \frac{1}{216} a^{3} + \frac{77}{216} a^{2} - \frac{5}{27}$, $\frac{1}{3456} a^{8} + \frac{1}{864} a^{6} - \frac{11}{1728} a^{5} - \frac{71}{864} a^{4} - \frac{211}{864} a^{3} - \frac{1}{4} a^{2} + \frac{31}{108} a + \frac{1}{3}$, $\frac{1}{124416} a^{9} - \frac{1}{10368} a^{7} - \frac{11}{62208} a^{6} - \frac{221}{10368} a^{5} + \frac{55}{10368} a^{4} + \frac{79}{3888} a^{3} + \frac{451}{1296} a^{2} - \frac{49}{243}$, $\frac{1}{497664} a^{10} + \frac{5}{41472} a^{8} - \frac{11}{248832} a^{7} - \frac{197}{41472} a^{6} - \frac{77}{41472} a^{5} + \frac{23}{486} a^{4} - \frac{523}{2592} a^{3} + \frac{1}{8} a^{2} + \frac{505}{1944} a + \frac{1}{3}$

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:  \( 814721686.599 \) (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 ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$