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Properties

Label 11.5.94156107000...0000.1
Degree $11$
Signature $[5, 3]$
Discriminant $-\,2^{9}\cdot 3^{14}\cdot 5^{6}\cdot 7^{5}\cdot 11^{4}$
Root discriminant $99.45$
Ramified primes $2, 3, 5, 7, 11$
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![-91, 203, 812, 375, -957, -1119, -351, 108, 63, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 14*x^9 + 63*x^8 + 108*x^7 - 351*x^6 - 1119*x^5 - 957*x^4 + 375*x^3 + 812*x^2 + 203*x - 91)
 
gp: K = bnfinit(x^11 - 2*x^10 - 14*x^9 + 63*x^8 + 108*x^7 - 351*x^6 - 1119*x^5 - 957*x^4 + 375*x^3 + 812*x^2 + 203*x - 91, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - 14 x^{9} + 63 x^{8} + 108 x^{7} - 351 x^{6} - 1119 x^{5} - 957 x^{4} + 375 x^{3} + 812 x^{2} + 203 x - 91 \)

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:  \(-9415610700088824000000=-\,2^{9}\cdot 3^{14}\cdot 5^{6}\cdot 7^{5}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.45$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{4}{27} a^{4} - \frac{4}{27} a^{3} - \frac{1}{3} a^{2} + \frac{13}{27} a - \frac{4}{27}$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{6} + \frac{1}{27} a^{5} - \frac{1}{9} a^{4} + \frac{2}{27} a^{3} - \frac{11}{27} a^{2} - \frac{2}{9} a - \frac{10}{27}$, $\frac{1}{81} a^{9} + \frac{1}{27} a^{6} - \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{4}{9} a + \frac{23}{81}$, $\frac{1}{58644} a^{10} - \frac{307}{58644} a^{9} + \frac{25}{6516} a^{8} + \frac{157}{9774} a^{7} - \frac{445}{9774} a^{6} - \frac{33}{724} a^{5} - \frac{421}{9774} a^{4} - \frac{2701}{19548} a^{3} - \frac{1445}{3258} a^{2} + \frac{859}{29322} a + \frac{27901}{58644}$

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:  \( 85295653.9896 \) (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 R R ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[3]$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.6.11.20$x^{6} + 24 x^{3} + 21$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$