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Properties

Label 11.3.15266822301...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{18}\cdot 3^{10}\cdot 5^{6}\cdot 631^{6}$
Root discriminant $683.73$
Ramified primes $2, 3, 5, 631$
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![-331642560, 221690590, -49404320, 5153040, -428892, 14304, -12042, 3495, 120, -84, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 84*x^9 + 120*x^8 + 3495*x^7 - 12042*x^6 + 14304*x^5 - 428892*x^4 + 5153040*x^3 - 49404320*x^2 + 221690590*x - 331642560)
 
gp: K = bnfinit(x^11 - 2*x^10 - 84*x^9 + 120*x^8 + 3495*x^7 - 12042*x^6 + 14304*x^5 - 428892*x^4 + 5153040*x^3 - 49404320*x^2 + 221690590*x - 331642560, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - 84 x^{9} + 120 x^{8} + 3495 x^{7} - 12042 x^{6} + 14304 x^{5} - 428892 x^{4} + 5153040 x^{3} - 49404320 x^{2} + 221690590 x - 331642560 \)

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:  \(15266822301029155208269824000000=2^{18}\cdot 3^{10}\cdot 5^{6}\cdot 631^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $683.73$
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:  $2, 3, 5, 631$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{5} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{1647} a^{6} + \frac{32}{1647} a^{5} - \frac{86}{1647} a^{4} - \frac{20}{1647} a^{3} - \frac{139}{1647} a^{2} - \frac{176}{1647} a + \frac{83}{549}$, $\frac{1}{4941} a^{7} + \frac{19}{549} a^{5} + \frac{170}{4941} a^{4} - \frac{199}{1647} a^{3} + \frac{43}{183} a^{2} - \frac{524}{4941} a + \frac{455}{1647}$, $\frac{1}{904203} a^{8} + \frac{85}{904203} a^{7} + \frac{1}{33489} a^{6} + \frac{1862}{904203} a^{5} - \frac{115}{904203} a^{4} + \frac{33842}{301401} a^{3} - \frac{206282}{904203} a^{2} - \frac{68888}{904203} a + \frac{39899}{301401}$, $\frac{1}{4521015} a^{9} + \frac{1}{4521015} a^{8} + \frac{23}{502335} a^{7} - \frac{406}{4521015} a^{6} - \frac{110407}{4521015} a^{5} + \frac{83483}{1507005} a^{4} - \frac{149413}{904203} a^{3} + \frac{329048}{904203} a^{2} - \frac{76289}{301401} a - \frac{16093}{33489}$, $\frac{1}{8273457450} a^{10} + \frac{28}{459636525} a^{9} + \frac{47}{91927305} a^{8} - \frac{1778}{30642435} a^{7} - \frac{151889}{551563830} a^{6} + \frac{6142177}{153212175} a^{5} + \frac{18932212}{1378909575} a^{4} + \frac{3194786}{55156383} a^{3} - \frac{25225034}{91927305} a^{2} - \frac{413247098}{827345745} a - \frac{131285083}{275781915}$

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:  \( 12101726961000 \) (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 R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 2 x^{4} + 6$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$$\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$$[5/4, 5/4]_{4}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
631Data not computed