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Properties

Label 12.0.55143062640...7504.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{18}\cdot 11^{18}$
Root discriminant $300.92$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![646853244, -41320152, -58182696, 2142756, 2387946, -281622, 5203, 1221, 1716, -407, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 407*x^9 + 1716*x^8 + 1221*x^7 + 5203*x^6 - 281622*x^5 + 2387946*x^4 + 2142756*x^3 - 58182696*x^2 - 41320152*x + 646853244)
 
gp: K = bnfinit(x^12 - 3*x^11 - 407*x^9 + 1716*x^8 + 1221*x^7 + 5203*x^6 - 281622*x^5 + 2387946*x^4 + 2142756*x^3 - 58182696*x^2 - 41320152*x + 646853244, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 407 x^{9} + 1716 x^{8} + 1221 x^{7} + 5203 x^{6} - 281622 x^{5} + 2387946 x^{4} + 2142756 x^{3} - 58182696 x^{2} - 41320152 x + 646853244 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(551430626404538014225974837504=2^{8}\cdot 3^{18}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $300.92$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{962} a^{10} + \frac{4}{481} a^{9} - \frac{147}{962} a^{8} + \frac{425}{962} a^{7} + \frac{51}{962} a^{6} + \frac{31}{962} a^{5} - \frac{94}{481} a^{4} - \frac{451}{962} a^{3} - \frac{17}{37} a^{2} + \frac{7}{481} a + \frac{77}{481}$, $\frac{1}{1030439370019761459910940151411981596} a^{11} - \frac{478511109073213347465197954943077}{1030439370019761459910940151411981596} a^{10} - \frac{44284078729327472228361243104565366}{257609842504940364977735037852995399} a^{9} + \frac{268805860183674576166093469520677305}{1030439370019761459910940151411981596} a^{8} + \frac{2719917031986971811686641105251237}{6962428175809199053452298320351227} a^{7} - \frac{220319248525895485690776812198059825}{1030439370019761459910940151411981596} a^{6} - \frac{38162365953438140473551797294149473}{1030439370019761459910940151411981596} a^{5} - \frac{182562308968884429100224350283508335}{515219685009880729955470075705990798} a^{4} + \frac{24371171551535489309666292867338729}{515219685009880729955470075705990798} a^{3} + \frac{208484113766928199032985913733810117}{515219685009880729955470075705990798} a^{2} - \frac{76783860235568269602387115244414324}{257609842504940364977735037852995399} a - \frac{2963348436007702586494381529400401}{257609842504940364977735037852995399}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{12}$, which has order $12$ (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:  $5$
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:  \( 3223036487.63 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,11)$ (as 12T179):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

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

Sibling fields

Degree 11 siblings: 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.4$x^{11} + 88 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$