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Properties

Label 11.11.1187692624...7424.1
Degree $11$
Signature $[11, 0]$
Discriminant $2^{24}\cdot 3^{10}\cdot 337^{4}\cdot 310501^{4}$
Root discriminant $10{,}157.60$
Ramified primes $2, 3, 337, 310501$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{11}$ (as 11T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3422014984884, -789230330881, 3011149862548, 177164975493, -53992425120, -3982035456, 88565592, 8989656, -17700, -5541, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 - 5541*x^9 - 17700*x^8 + 8989656*x^7 + 88565592*x^6 - 3982035456*x^5 - 53992425120*x^4 + 177164975493*x^3 + 3011149862548*x^2 - 789230330881*x - 3422014984884)
 
gp: K = bnfinit(x^11 - 4*x^10 - 5541*x^9 - 17700*x^8 + 8989656*x^7 + 88565592*x^6 - 3982035456*x^5 - 53992425120*x^4 + 177164975493*x^3 + 3011149862548*x^2 - 789230330881*x - 3422014984884, 1)
 

Normalized defining polynomial

\( x^{11} - 4 x^{10} - 5541 x^{9} - 17700 x^{8} + 8989656 x^{7} + 88565592 x^{6} - 3982035456 x^{5} - 53992425120 x^{4} + 177164975493 x^{3} + 3011149862548 x^{2} - 789230330881 x - 3422014984884 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(118769262421915560193703211428553337469927424=2^{24}\cdot 3^{10}\cdot 337^{4}\cdot 310501^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10{,}157.60$
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:  $2, 3, 337, 310501$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{10} - \frac{2950442336895763005424408194123522436283571976756718491285}{57887320607168480035725746532474501690526662399435613406516} a^{9} + \frac{4431843270042285766136492654584063718964881085264741564690}{14471830151792120008931436633118625422631665599858903351629} a^{8} - \frac{559722587760387350640283751740839727238365608050464518991}{14471830151792120008931436633118625422631665599858903351629} a^{7} + \frac{505295161275915918479521011154303348167337535019698801991}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{1556176856207666021957244551999529442651090629791143122428}{14471830151792120008931436633118625422631665599858903351629} a^{5} + \frac{3154123839974750402397416329553678833755212782357510662549}{14471830151792120008931436633118625422631665599858903351629} a^{4} + \frac{4556568892170348868244497265034681109791694349187835610075}{14471830151792120008931436633118625422631665599858903351629} a^{3} + \frac{28310274281757867439265575523407866221021418921950949260633}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{19499521295137430063261590875714689539962535653858132127485}{57887320607168480035725746532474501690526662399435613406516} a - \frac{3057362392104798823686377905115400888834880911517503355097}{14471830151792120008931436633118625422631665599858903351629}$

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:  $10$
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:  \( 71167879092500000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 11T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

Intermediate fields

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.24.66$x^{8} + 20 x^{4} + 52$$8$$1$$24$$QD_{16}$$[2, 3, 4]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
337Data not computed
310501Data not computed