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Properties

Label 11.3.74136784216...9776.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{8}\cdot 3^{16}\cdot 11^{20}$
Root discriminant $640.28$
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![-631310436, 177408, 332288, 354816, 234432, 96096, 22176, 1584, -462, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 99*x^9 - 462*x^8 + 1584*x^7 + 22176*x^6 + 96096*x^5 + 234432*x^4 + 354816*x^3 + 332288*x^2 + 177408*x - 631310436)
 
gp: K = bnfinit(x^11 - 99*x^9 - 462*x^8 + 1584*x^7 + 22176*x^6 + 96096*x^5 + 234432*x^4 + 354816*x^3 + 332288*x^2 + 177408*x - 631310436, 1)
 

Normalized defining polynomial

\( x^{11} - 99 x^{9} - 462 x^{8} + 1584 x^{7} + 22176 x^{6} + 96096 x^{5} + 234432 x^{4} + 354816 x^{3} + 332288 x^{2} + 177408 x - 631310436 \)

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:  \(7413678421661011080149217259776=2^{8}\cdot 3^{16}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $640.28$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{3} + \frac{4}{27} a^{2} + \frac{13}{27} a - \frac{1}{3}$, $\frac{1}{243} a^{4} - \frac{1}{81} a^{3} + \frac{13}{81} a^{2} + \frac{62}{243} a - \frac{8}{27}$, $\frac{1}{2187} a^{5} - \frac{1}{2187} a^{4} + \frac{11}{729} a^{3} - \frac{346}{2187} a^{2} - \frac{191}{2187} a - \frac{97}{243}$, $\frac{1}{4374} a^{6} - \frac{1}{4374} a^{5} + \frac{1}{729} a^{4} - \frac{11}{2187} a^{3} - \frac{136}{2187} a^{2} + \frac{34}{243} a + \frac{4}{9}$, $\frac{1}{13122} a^{7} + \frac{1}{13122} a^{6} - \frac{1}{6561} a^{5} - \frac{2}{6561} a^{4} - \frac{14}{6561} a^{3} - \frac{143}{6561} a^{2} + \frac{1043}{2187} a + \frac{7}{243}$, $\frac{1}{118098} a^{8} + \frac{1}{39366} a^{7} - \frac{2}{19683} a^{6} - \frac{10}{59049} a^{5} + \frac{13}{19683} a^{4} + \frac{260}{19683} a^{3} + \frac{4496}{59049} a^{2} - \frac{1147}{6561} a + \frac{323}{729}$, $\frac{1}{1062882} a^{9} - \frac{2}{531441} a^{8} - \frac{1}{177147} a^{7} - \frac{17}{1062882} a^{6} + \frac{109}{531441} a^{5} + \frac{295}{177147} a^{4} + \frac{3329}{531441} a^{3} - \frac{26000}{531441} a^{2} - \frac{20765}{59049} a - \frac{3122}{6561}$, $\frac{1}{76527504} a^{10} + \frac{7}{76527504} a^{9} - \frac{25}{38263752} a^{8} + \frac{283}{19131876} a^{7} - \frac{539}{19131876} a^{6} - \frac{4061}{19131876} a^{5} - \frac{27245}{19131876} a^{4} - \frac{269159}{19131876} a^{3} + \frac{1245979}{19131876} a^{2} - \frac{61175}{2125764} a - \frac{37961}{236196}$

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:  \( 2156019271050 \) (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 ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.15.20$x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{6} + 3 x^{3} + 6$$9$$1$$15$$C_3^2 : D_{6} $$[3/2, 3/2, 2]_{2}^{2}$
$11$11.11.20.13$x^{11} + 66 x^{10} + 11$$11$$1$$20$$C_{11}:C_5$$[2]^{5}$