↑

Properties

Label 12.0.56466496143...4096.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 11^{18}$
Root discriminant $536.19$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2961900172, 428379792, 85826268, 97123400, 16259463, -967824, 1098504, -9504, -1683, -2332, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 66*x^10 - 2332*x^9 - 1683*x^8 - 9504*x^7 + 1098504*x^6 - 967824*x^5 + 16259463*x^4 + 97123400*x^3 + 85826268*x^2 + 428379792*x + 2961900172)
 
gp: K = bnfinit(x^12 + 66*x^10 - 2332*x^9 - 1683*x^8 - 9504*x^7 + 1098504*x^6 - 967824*x^5 + 16259463*x^4 + 97123400*x^3 + 85826268*x^2 + 428379792*x + 2961900172, 1)
 

Normalized defining polynomial

\( x^{12} + 66 x^{10} - 2332 x^{9} - 1683 x^{8} - 9504 x^{7} + 1098504 x^{6} - 967824 x^{5} + 16259463 x^{4} + 97123400 x^{3} + 85826268 x^{2} + 428379792 x + 2961900172 \)

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:  \(564664961438246926567398233604096=2^{18}\cdot 3^{18}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $536.19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} - \frac{1}{3} a^{4} + \frac{1}{4} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} + \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{1}{18}$, $\frac{1}{72} a^{10} - \frac{1}{24} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} + \frac{3}{8} a^{2} + \frac{4}{9} a + \frac{1}{12}$, $\frac{1}{219179927144267973718124032670747068526544} a^{11} + \frac{21304538845796387974775761053699132038}{4566248482172249452460917347307230594303} a^{10} + \frac{546286295756946364701592069808640754883}{54794981786066993429531008167686767131636} a^{9} + \frac{84619544598103062743372473274547902985}{6088331309562999269947889796409640792404} a^{8} + \frac{2939172215698634705008171212858945172403}{73059975714755991239374677556915689508848} a^{7} + \frac{1039322219969749801934073910752914415407}{18264993928688997809843669389228922377212} a^{6} - \frac{1536208506590707833592664991730054954261}{36529987857377995619687338778457844754424} a^{5} - \frac{3712156609247957747345528916162617444293}{18264993928688997809843669389228922377212} a^{4} - \frac{29568907355964137598556276610380151461847}{73059975714755991239374677556915689508848} a^{3} - \frac{1227434417458684639952199640622320393727}{3223234222709823142913588715746280419508} a^{2} - \frac{323785522014500528047013740048289310253}{716274271713294031758575270165840093224} a - \frac{4793595567496863411119070155706066405307}{27397490893033496714765504083843383565818}$

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

Class group and class number

$C_{2}\times C_{6}$, 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:  \( 177176668782 \) (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 ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.2$x^{11} + 77 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$