↑

Properties

Label 12.0.55143062640...7504.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{18}\cdot 11^{18}$
Root discriminant $300.92$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[12]$ (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![214378768, -66441360, -21351792, 6173992, 1995840, -483912, -28215, -6435, 4950, -143, -66, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 66*x^10 - 143*x^9 + 4950*x^8 - 6435*x^7 - 28215*x^6 - 483912*x^5 + 1995840*x^4 + 6173992*x^3 - 21351792*x^2 - 66441360*x + 214378768)
 
gp: K = bnfinit(x^12 - 3*x^11 - 66*x^10 - 143*x^9 + 4950*x^8 - 6435*x^7 - 28215*x^6 - 483912*x^5 + 1995840*x^4 + 6173992*x^3 - 21351792*x^2 - 66441360*x + 214378768, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 66 x^{10} - 143 x^{9} + 4950 x^{8} - 6435 x^{7} - 28215 x^{6} - 483912 x^{5} + 1995840 x^{4} + 6173992 x^{3} - 21351792 x^{2} - 66441360 x + 214378768 \)

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:  \(551430626404538014225974837504=2^{8}\cdot 3^{18}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $300.92$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{144} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{7} - \frac{1}{48} a^{6} - \frac{1}{24} a^{5} - \frac{1}{48} a^{4} - \frac{1}{48} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{11}{36}$, $\frac{1}{288} a^{10} - \frac{1}{288} a^{9} - \frac{1}{24} a^{8} - \frac{1}{96} a^{7} + \frac{1}{24} a^{6} + \frac{7}{96} a^{5} + \frac{5}{96} a^{4} + \frac{17}{48} a^{3} + \frac{1}{3} a^{2} + \frac{25}{72} a - \frac{17}{36}$, $\frac{1}{266961331222886562558460055136} a^{11} + \frac{183902324534308043333565551}{266961331222886562558460055136} a^{10} + \frac{460651122410926912052456779}{133480665611443281279230027568} a^{9} + \frac{3260349480080708183519954269}{88987110407628854186153351712} a^{8} - \frac{2327663358834178447182533}{397263885748343099045327463} a^{7} - \frac{691169878378237620521218361}{29662370135876284728717783904} a^{6} - \frac{175142859622715314340139571}{4237481447982326389816826272} a^{5} - \frac{191490944562170210216651492}{926949066746133897772430747} a^{4} - \frac{1008786092840057674955057873}{44493555203814427093076675856} a^{3} - \frac{23595220827347697945625904303}{66740332805721640639615013784} a^{2} + \frac{1533516017953071572383699411}{33370166402860820319807506892} a - \frac{1709914350901282518021310199}{33370166402860820319807506892}$

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

Class group and class number

$C_{12}$, 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:  \( 5472245079.5 \) (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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[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}$