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Properties

Label 12.8.94768777961...7264.1
Degree $12$
Signature $[8, 2]$
Discriminant $2^{14}\cdot 3^{16}\cdot 11591846801^{2}$
Root discriminant $462.09$
Ramified primes $2, 3, 11591846801$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{12}$ (as 12T300)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21, 254, 191, -2170, -3840, 48, 2604, 384, -480, 30, 43, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 43*x^10 + 30*x^9 - 480*x^8 + 384*x^7 + 2604*x^6 + 48*x^5 - 3840*x^4 - 2170*x^3 + 191*x^2 + 254*x + 21)
 
gp: K = bnfinit(x^12 - 2*x^11 + 43*x^10 + 30*x^9 - 480*x^8 + 384*x^7 + 2604*x^6 + 48*x^5 - 3840*x^4 - 2170*x^3 + 191*x^2 + 254*x + 21, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 43 x^{10} + 30 x^{9} - 480 x^{8} + 384 x^{7} + 2604 x^{6} + 48 x^{5} - 3840 x^{4} - 2170 x^{3} + 191 x^{2} + 254 x + 21 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(94768777961133151657570669707264=2^{14}\cdot 3^{16}\cdot 11591846801^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.09$
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:  $2, 3, 11591846801$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{27} a^{7} + \frac{4}{27} a^{4} + \frac{1}{9} a^{3} - \frac{5}{27} a - \frac{1}{9}$, $\frac{1}{27} a^{8} + \frac{1}{27} a^{5} - \frac{1}{9} a^{4} + \frac{7}{27} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{81} a^{9} + \frac{1}{81} a^{8} + \frac{1}{81} a^{7} + \frac{4}{81} a^{6} - \frac{2}{81} a^{5} - \frac{11}{81} a^{4} - \frac{2}{81} a^{3} + \frac{1}{81} a^{2} + \frac{10}{81} a - \frac{1}{27}$, $\frac{1}{162} a^{10} - \frac{1}{27} a^{6} - \frac{2}{27} a^{4} - \frac{1}{27} a^{3} + \frac{1}{3} a^{2} - \frac{8}{81} a - \frac{5}{54}$, $\frac{1}{972} a^{11} - \frac{1}{972} a^{10} - \frac{1}{162} a^{8} - \frac{1}{54} a^{7} - \frac{5}{162} a^{6} - \frac{1}{54} a^{5} - \frac{7}{162} a^{4} + \frac{1}{162} a^{3} + \frac{26}{243} a^{2} - \frac{101}{972} a + \frac{35}{324}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 3786617937910 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{12}$ (as 12T300):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 239500800
The 43 conjugacy class representatives for $A_{12}$
Character table for $A_{12}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.15.33$x^{9} + 6 x^{7} + 3 x^{6} + 3$$9$$1$$15$$(C_3^2:C_8):C_2$$[15/8, 15/8]_{8}^{2}$
11591846801Data not computed