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Properties

Label 11.11.1305275211...0000.1
Degree $11$
Signature $[11, 0]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{6}\cdot 11^{12}\cdot 13^{4}\cdot 17^{4}\cdot 347^{8}$
Root discriminant $67{,}406.25$
Ramified primes $2, 3, 5, 11, 13, 17, 347$
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![-432518584183431877728, -26705329205451737544, 562256553231789768, 38125562570421876, -142840655647272, -14713278229518, 17620911770, 1999274189, -1016268, -89045, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 89045*x^9 - 1016268*x^8 + 1999274189*x^7 + 17620911770*x^6 - 14713278229518*x^5 - 142840655647272*x^4 + 38125562570421876*x^3 + 562256553231789768*x^2 - 26705329205451737544*x - 432518584183431877728)
 
gp: K = bnfinit(x^11 - 89045*x^9 - 1016268*x^8 + 1999274189*x^7 + 17620911770*x^6 - 14713278229518*x^5 - 142840655647272*x^4 + 38125562570421876*x^3 + 562256553231789768*x^2 - 26705329205451737544*x - 432518584183431877728, 1)
 

Normalized defining polynomial

\( x^{11} - 89045 x^{9} - 1016268 x^{8} + 1999274189 x^{7} + 17620911770 x^{6} - 14713278229518 x^{5} - 142840655647272 x^{4} + 38125562570421876 x^{3} + 562256553231789768 x^{2} - 26705329205451737544 x - 432518584183431877728 \)

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:  \(130527521176011960658406976951299254564347331584000000=2^{16}\cdot 3^{4}\cdot 5^{6}\cdot 11^{12}\cdot 13^{4}\cdot 17^{4}\cdot 347^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67{,}406.25$
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:  $2, 3, 5, 11, 13, 17, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{36} a^{7} - \frac{5}{36} a^{5} + \frac{17}{36} a^{3} - \frac{5}{18} a^{2} + \frac{1}{6} a$, $\frac{1}{648} a^{8} + \frac{1}{108} a^{7} + \frac{19}{648} a^{6} - \frac{5}{36} a^{5} + \frac{41}{648} a^{4} + \frac{37}{81} a^{3} + \frac{7}{108} a^{2} + \frac{7}{18} a - \frac{1}{3}$, $\frac{1}{7776} a^{9} - \frac{1}{1296} a^{8} + \frac{1}{7776} a^{7} - \frac{53}{1296} a^{6} - \frac{13}{7776} a^{5} + \frac{59}{1944} a^{4} - \frac{25}{54} a^{2} - \frac{11}{24} a - \frac{1}{6}$, $\frac{1}{77090518032876356986361559183128194281316998903760442131223557956419187462048} a^{10} + \frac{124063824109717207811410820816235999591526460332528610331902001644178931}{6424209836073029748863463265260682856776416575313370177601963163034932288504} a^{9} - \frac{30597807054012888555543053361993098289895970225172927598092080875412361839}{77090518032876356986361559183128194281316998903760442131223557956419187462048} a^{8} + \frac{42592973244605563039613474060573141846843421147995757643993945015708696547}{6424209836073029748863463265260682856776416575313370177601963163034932288504} a^{7} - \frac{3397979810222696923522279559141674664025155511576946326608907042382619090629}{77090518032876356986361559183128194281316998903760442131223557956419187462048} a^{6} - \frac{4491702218155769171317456389211903186928899129985277764783598497025513890159}{38545259016438178493180779591564097140658499451880221065611778978209593731024} a^{5} + \frac{299148794936799968956450836957272292047159246211406195311013301522315106125}{2141403278691009916287821088420227618925472191771123392533987721011644096168} a^{4} - \frac{1061960809164069760108824405780150496372038977921181730234038934933019189}{3304634689337978265876267111759610523033136098412227457614178581808092741} a^{3} + \frac{321158168580254997512153695310359398427903871437781495873542425398227646989}{713801092897003305429273696140075872975157397257041130844662573670548032056} a^{2} + \frac{56136894784153294320012583393770847842758659479514119210015353338553716385}{118966848816167217571545616023345978829192899542840188474110428945091338676} a + \frac{244584859911029288428486644452659479364159664353985251664878389434216105}{583170827530231458684047137369343033476435782072746021931913867377898719}$

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:  \( 6854659729190000000000000 \) (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 ${\href{/LocalNumberField/7.11.0.1}{11} }$ R R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$11$11.11.12.4$x^{11} + 110 x^{2} + 11$$11$$1$$12$$C_{11}:C_5$$[6/5]_{5}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
347Data not computed