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Properties

Label 11.1.16198260678...0000.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,2^{21}\cdot 5^{8}\cdot 7^{11}$
Root discriminant $84.75$
Ramified primes $2, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (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![200, 1080, -1024, 384, -384, 224, 56, -24, 9, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 9*x^9 + 9*x^8 - 24*x^7 + 56*x^6 + 224*x^5 - 384*x^4 + 384*x^3 - 1024*x^2 + 1080*x + 200)
 
gp: K = bnfinit(x^11 - x^10 - 9*x^9 + 9*x^8 - 24*x^7 + 56*x^6 + 224*x^5 - 384*x^4 + 384*x^3 - 1024*x^2 + 1080*x + 200, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 9 x^{9} + 9 x^{8} - 24 x^{7} + 56 x^{6} + 224 x^{5} - 384 x^{4} + 384 x^{3} - 1024 x^{2} + 1080 x + 200 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1619826067865600000000=-\,2^{21}\cdot 5^{8}\cdot 7^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.75$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{6} - \frac{1}{2} a^{3} - \frac{2}{5} a$, $\frac{1}{10} a^{7} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{20} a^{8} - \frac{1}{20} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{20} a^{9} - \frac{1}{20} a^{7} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{2494040} a^{10} - \frac{2213}{124702} a^{9} - \frac{40387}{2494040} a^{8} - \frac{14366}{311755} a^{7} - \frac{60477}{1247020} a^{6} - \frac{65978}{311755} a^{5} + \frac{26849}{124702} a^{4} - \frac{247463}{623510} a^{3} + \frac{100054}{311755} a^{2} - \frac{59743}{311755} a + \frac{9343}{62351}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $5$
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:  \( 8618230.56599 \) (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 ${\href{/LocalNumberField/3.11.0.1}{11} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.21.3$x^{8} + 10 x^{6} + 2$$8$$1$$21$$C_2 \wr C_2\wr C_2$$[2, 2, 3, 7/2, 7/2, 15/4]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.9.8.1$x^{9} - 5$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.7.9.2$x^{7} + 7 x^{3} + 7$$7$$1$$9$$F_7$$[3/2]_{2}^{3}$