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Properties

Label 12.0.26562918623...6864.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 3^{6}\cdot 11^{18}$
Root discriminant $159.23$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![2459241, -803034, -350757, -19602, 141768, -4158, 3003, 1320, -176, 88, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 11*x^10 + 88*x^9 - 176*x^8 + 1320*x^7 + 3003*x^6 - 4158*x^5 + 141768*x^4 - 19602*x^3 - 350757*x^2 - 803034*x + 2459241)
 
gp: K = bnfinit(x^12 - 4*x^11 + 11*x^10 + 88*x^9 - 176*x^8 + 1320*x^7 + 3003*x^6 - 4158*x^5 + 141768*x^4 - 19602*x^3 - 350757*x^2 - 803034*x + 2459241, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 11 x^{10} + 88 x^{9} - 176 x^{8} + 1320 x^{7} + 3003 x^{6} - 4158 x^{5} + 141768 x^{4} - 19602 x^{3} - 350757 x^{2} - 803034 x + 2459241 \)

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:  \(265629186230572597224996864=2^{16}\cdot 3^{6}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $159.23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{4}{27} a^{7} - \frac{2}{27} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{5198483242700165697008539149} a^{11} - \frac{42537375807552739778519023}{5198483242700165697008539149} a^{10} + \frac{283885760725667008235626709}{5198483242700165697008539149} a^{9} + \frac{83914992064584138934678693}{5198483242700165697008539149} a^{8} + \frac{731779193004803502127627585}{5198483242700165697008539149} a^{7} - \frac{45463497178476584520267160}{577609249188907299667615461} a^{6} - \frac{318971425048035805229423456}{1732827747566721899002846383} a^{5} - \frac{23147298469967049739952861}{192536416396302433222538487} a^{4} + \frac{55854657017800877887844947}{577609249188907299667615461} a^{3} + \frac{361333755992438108780583}{64178805465434144407512829} a^{2} + \frac{56408517115934902212763423}{192536416396302433222538487} a + \frac{26493810755172609310737675}{64178805465434144407512829}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 68105787.4587 \) (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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$