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Properties

Label 12.0.22239669253...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $278.99$
Ramified primes $2, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (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![3509901, 10626358, 13245771, 8129220, 2461635, 325138, 9394, -9922, 55, 440, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 11*x^10 + 440*x^9 + 55*x^8 - 9922*x^7 + 9394*x^6 + 325138*x^5 + 2461635*x^4 + 8129220*x^3 + 13245771*x^2 + 10626358*x + 3509901)
 
gp: K = bnfinit(x^12 - 2*x^11 + 11*x^10 + 440*x^9 + 55*x^8 - 9922*x^7 + 9394*x^6 + 325138*x^5 + 2461635*x^4 + 8129220*x^3 + 13245771*x^2 + 10626358*x + 3509901, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 11 x^{10} + 440 x^{9} + 55 x^{8} - 9922 x^{7} + 9394 x^{6} + 325138 x^{5} + 2461635 x^{4} + 8129220 x^{3} + 13245771 x^{2} + 10626358 x + 3509901 \)

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:  \(222396692539689259240000000000=2^{12}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.99$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{20} a^{8} - \frac{1}{20} a^{6} + \frac{1}{5} a^{3} + \frac{1}{4} a^{2} - \frac{1}{5} a - \frac{1}{4}$, $\frac{1}{20} a^{9} - \frac{1}{20} a^{7} + \frac{1}{5} a^{4} + \frac{1}{4} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3760} a^{10} - \frac{37}{1880} a^{9} - \frac{41}{1880} a^{8} - \frac{43}{1880} a^{7} - \frac{11}{752} a^{6} + \frac{1}{188} a^{5} - \frac{331}{3760} a^{4} + \frac{331}{1880} a^{3} - \frac{687}{1880} a^{2} + \frac{111}{376} a + \frac{1609}{3760}$, $\frac{1}{959187672124539774894845600} a^{11} - \frac{28881346925205277002851}{959187672124539774894845600} a^{10} + \frac{443868772758922370507581}{47959383606226988744742280} a^{9} - \frac{361265671217226483071803}{23979691803113494372371140} a^{8} - \frac{2597414391687540594931809}{191837534424907954978969120} a^{7} + \frac{40184118022722178922138063}{959187672124539774894845600} a^{6} - \frac{146394693403850728511373343}{959187672124539774894845600} a^{5} + \frac{55244558618594495211574581}{191837534424907954978969120} a^{4} - \frac{6175035483230454581662267}{23979691803113494372371140} a^{3} - \frac{118872782303726589427763}{255103104288441429493310} a^{2} - \frac{6192352482701143957985089}{959187672124539774894845600} a - \frac{150117137738807288173396981}{959187672124539774894845600}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 17444898307.7 \) (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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.11.10.1$x^{11} - 5$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$$\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}$