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Properties

Label 12.0.23319983427...4224.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 11^{18}$
Root discriminant $130.01$
Ramified primes $2, 11$
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![1138599, -1124904, 760364, -305404, -9317, 31944, -4840, 3872, 253, 0, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 44*x^10 + 253*x^8 + 3872*x^7 - 4840*x^6 + 31944*x^5 - 9317*x^4 - 305404*x^3 + 760364*x^2 - 1124904*x + 1138599)
 
gp: K = bnfinit(x^12 - 4*x^11 + 44*x^10 + 253*x^8 + 3872*x^7 - 4840*x^6 + 31944*x^5 - 9317*x^4 - 305404*x^3 + 760364*x^2 - 1124904*x + 1138599, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 44 x^{10} + 253 x^{8} + 3872 x^{7} - 4840 x^{6} + 31944 x^{5} - 9317 x^{4} - 305404 x^{3} + 760364 x^{2} - 1124904 x + 1138599 \)

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:  \(23319983427649720469684224=2^{22}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.01$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{3}{8}$, $\frac{1}{24} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} + \frac{5}{24} a - \frac{1}{2}$, $\frac{1}{24} a^{10} - \frac{1}{4} a^{4} + \frac{11}{24} a^{2} - \frac{1}{4}$, $\frac{1}{570399883930437421597104} a^{11} + \frac{10363046439901437990269}{570399883930437421597104} a^{10} - \frac{8359542577205341621589}{570399883930437421597104} a^{9} - \frac{3092368699485073584889}{570399883930437421597104} a^{8} + \frac{3408285204998251971513}{95066647321739570266184} a^{7} - \frac{30481780322762858381405}{285199941965218710798552} a^{6} - \frac{31904694465999174687577}{285199941965218710798552} a^{5} - \frac{61008159080649634595125}{285199941965218710798552} a^{4} - \frac{59618349761480165666623}{570399883930437421597104} a^{3} + \frac{134907184681804971550709}{570399883930437421597104} a^{2} + \frac{3146979953324847549823}{570399883930437421597104} a + \frac{46236071376466340665945}{190133294643479140532368}$

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:  \( 103922902.785 \) (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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.1$x^{11} + 66 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$