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Properties

Label 11.5.84728860944...0000.1
Degree $11$
Signature $[5, 3]$
Discriminant $-\,2^{8}\cdot 3^{25}\cdot 5^{8}$
Root discriminant $64.81$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_{11}$ (as 11T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-324, 324, 405, -405, 0, 0, 0, 0, 0, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324)
 
gp: K = bnfinit(x^11 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} + 10 x^{9} - 405 x^{3} + 405 x^{2} + 324 x - 324 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-84728860944300000000=-\,2^{8}\cdot 3^{25}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.81$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{5}$, $\frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{1}{27} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{243540} a^{10} + \frac{1073}{243540} a^{9} - \frac{2792}{60885} a^{8} - \frac{571}{20295} a^{7} - \frac{2149}{20295} a^{6} + \frac{1094}{20295} a^{5} - \frac{991}{2255} a^{4} - \frac{2177}{6765} a^{3} + \frac{3441}{9020} a^{2} - \frac{2551}{9020} a + \frac{1159}{4510}$

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:  $7$
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:  \( 5378394.89404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{11}$ (as 11T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 39916800
The 56 conjugacy class representatives for $S_{11}$ are not computed
Character table for $S_{11}$ is not computed

Intermediate fields

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

Sibling fields

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 ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.24.21$x^{9} + 18 x^{8} + 9 x^{7} + 6$$9$$1$$24$$(C_3^3:C_3):C_2$$[3/2, 5/2, 8/3, 19/6]_{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$