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Properties

Label 12.0.17000267918...9296.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 3^{6}\cdot 11^{18}$
Root discriminant $225.18$
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![249705, -334404, -37818, 150876, -27005, -15488, 38104, 23848, 4455, -220, -110, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 110*x^10 - 220*x^9 + 4455*x^8 + 23848*x^7 + 38104*x^6 - 15488*x^5 - 27005*x^4 + 150876*x^3 - 37818*x^2 - 334404*x + 249705)
 
gp: K = bnfinit(x^12 - 4*x^11 - 110*x^10 - 220*x^9 + 4455*x^8 + 23848*x^7 + 38104*x^6 - 15488*x^5 - 27005*x^4 + 150876*x^3 - 37818*x^2 - 334404*x + 249705, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 110 x^{10} - 220 x^{9} + 4455 x^{8} + 23848 x^{7} + 38104 x^{6} - 15488 x^{5} - 27005 x^{4} + 150876 x^{3} - 37818 x^{2} - 334404 x + 249705 \)

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:  \(17000267918756646222399799296=2^{22}\cdot 3^{6}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{7} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{7}{16} a + \frac{5}{16}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} + \frac{1}{24} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{12} a^{3} - \frac{11}{48} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{290533132396167193203792} a^{11} + \frac{238199610903744439891}{26412102945106108473072} a^{10} - \frac{294921850075452645193}{26412102945106108473072} a^{9} + \frac{524194503801955494967}{26412102945106108473072} a^{8} + \frac{26467096794013116289}{1100504289379421186378} a^{7} - \frac{579136536949708219475}{13206051472553054236536} a^{6} - \frac{5719143231286239608453}{13206051472553054236536} a^{5} - \frac{3117479277112697021197}{6603025736276527118268} a^{4} - \frac{305522007234636008083}{26412102945106108473072} a^{3} - \frac{3346935915396195041423}{8804034315035369491024} a^{2} - \frac{1740475319495452221939}{8804034315035369491024} a + \frac{10081430827645719397829}{96844377465389064401264}$

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:  \( 1201113875.2 \) (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 ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$$[3]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$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$$0$Trivial$[\ ]$
11.11.18.5$x^{11} + 110 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$