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Properties

Label 11.3.25351957617...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{18}\cdot 3^{10}\cdot 5^{10}\cdot 109^{6}$
Root discriminant $471.08$
Ramified primes $2, 3, 5, 109$
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![-41776622, 27421202, -6637580, 973740, -12180, -30408, -8412, 3480, 135, -95, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 95*x^9 + 135*x^8 + 3480*x^7 - 8412*x^6 - 30408*x^5 - 12180*x^4 + 973740*x^3 - 6637580*x^2 + 27421202*x - 41776622)
 
gp: K = bnfinit(x^11 - x^10 - 95*x^9 + 135*x^8 + 3480*x^7 - 8412*x^6 - 30408*x^5 - 12180*x^4 + 973740*x^3 - 6637580*x^2 + 27421202*x - 41776622, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 95 x^{9} + 135 x^{8} + 3480 x^{7} - 8412 x^{6} - 30408 x^{5} - 12180 x^{4} + 973740 x^{3} - 6637580 x^{2} + 27421202 x - 41776622 \)

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:  \(253519576179328535040000000000=2^{18}\cdot 3^{10}\cdot 5^{10}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $471.08$
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:  $2, 3, 5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{509} a^{6} + \frac{82}{509} a^{5} - \frac{212}{509} a^{4} + \frac{136}{509} a^{3} + \frac{170}{509} a^{2} + \frac{149}{509} a - \frac{124}{509}$, $\frac{1}{509} a^{7} + \frac{190}{509} a^{5} + \frac{214}{509} a^{4} + \frac{216}{509} a^{3} - \frac{48}{509} a^{2} - \frac{126}{509} a - \frac{12}{509}$, $\frac{1}{259081} a^{8} + \frac{227}{259081} a^{7} + \frac{177}{259081} a^{6} + \frac{9193}{259081} a^{5} + \frac{61221}{259081} a^{4} - \frac{17936}{259081} a^{3} + \frac{74316}{259081} a^{2} - \frac{17826}{259081} a + \frac{10595}{259081}$, $\frac{1}{777243} a^{9} + \frac{19}{259081} a^{7} + \frac{21}{259081} a^{6} - \frac{25034}{259081} a^{5} + \frac{116371}{259081} a^{4} + \frac{13907}{259081} a^{3} - \frac{115189}{259081} a^{2} - \frac{41955}{259081} a + \frac{382219}{777243}$, $\frac{1}{192269709882} a^{10} + \frac{29708}{96134854941} a^{9} + \frac{15341}{64089903294} a^{8} - \frac{17468341}{32044951647} a^{7} - \frac{12540679}{32044951647} a^{6} - \frac{191069869}{395616687} a^{5} - \frac{8917872436}{32044951647} a^{4} - \frac{13587570184}{32044951647} a^{3} - \frac{2976663940}{10681650549} a^{2} - \frac{23478327733}{96134854941} a - \frac{47296701463}{96134854941}$

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:  \( 300300836741 \) (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 ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.16.71$x^{8} + 6 x^{6} + 20$$8$$1$$16$$\textrm{GL(2,3)}$$[4/3, 4/3, 3]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.9.12$x^{9} + 6 x + 6$$9$$1$$9$$(C_3^2:C_8):C_2$$[9/8, 9/8]_{8}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.8.6.4$x^{8} + 1853 x^{4} + 2566296$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$