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Properties

Label 11.3.65610000000...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{18}\cdot 3^{8}\cdot 5^{18}$
Root discriminant $96.24$
Ramified primes $2, 3, 5$
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![90, 2805, -10950, -5075, -1400, 796, 232, 70, -50, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90)
 
gp: K = bnfinit(x^11 - 2*x^10 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90 \)

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:  \(6561000000000000000000=2^{18}\cdot 3^{8}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1894046861953608} a^{10} + \frac{23569802922049}{1894046861953608} a^{9} + \frac{19593327064199}{947023430976804} a^{8} - \frac{97955798617415}{473511715488402} a^{7} + \frac{302323532157431}{947023430976804} a^{6} + \frac{196073990354663}{947023430976804} a^{5} - \frac{422970158627623}{947023430976804} a^{4} - \frac{307579548304303}{947023430976804} a^{3} - \frac{800733488315849}{1894046861953608} a^{2} + \frac{40061575935705}{631348953984536} a + \frac{49459480584111}{315674476992268}$

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:  \( 107079411.141 \) (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 ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$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$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 10.2e18_3e8_5e18.11t6.1c1$10$ $ 2^{18} \cdot 3^{8} \cdot 5^{18}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $1$ $2$
10.2e24_3e9_5e20.330.1c1$10$ $ 2^{24} \cdot 3^{9} \cdot 5^{20}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $0$ $-2$
10.2e24_3e9_5e20.330.1c2$10$ $ 2^{24} \cdot 3^{9} \cdot 5^{20}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $0$ $-2$
11.2e18_3e10_5e20.12t272.1c1$11$ $ 2^{18} \cdot 3^{10} \cdot 5^{20}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $1$ $3$
16.2e36_3e14_5e30.144.1c1$16$ $ 2^{36} \cdot 3^{14} \cdot 5^{30}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $0$ $0$
16.2e36_3e14_5e30.144.1c2$16$ $ 2^{36} \cdot 3^{14} \cdot 5^{30}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $0$ $0$
44.2e90_3e38_5e86.55.1c1$44$ $ 2^{90} \cdot 3^{38} \cdot 5^{86}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $1$ $4$
45.2e102_3e40_5e88.110.1c1$45$ $ 2^{102} \cdot 3^{40} \cdot 5^{88}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $1$ $-3$
55.2e120_3e48_5e108.110.1c1$55$ $ 2^{120} \cdot 3^{48} \cdot 5^{108}$ $x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$ $M_{11}$ (as 11T6) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.