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Properties

Label 12.0.18260173718028288.64
Degree $12$
Signature $[0, 6]$
Discriminant $2^{35}\cdot 3^{12}$
Root discriminant $22.65$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3:S_3.C_2$ (as 12T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![98, 336, 396, 104, -183, -216, -64, 72, 54, -8, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 8*x^9 + 54*x^8 + 72*x^7 - 64*x^6 - 216*x^5 - 183*x^4 + 104*x^3 + 396*x^2 + 336*x + 98)
 
gp: K = bnfinit(x^12 - 12*x^10 - 8*x^9 + 54*x^8 + 72*x^7 - 64*x^6 - 216*x^5 - 183*x^4 + 104*x^3 + 396*x^2 + 336*x + 98, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} - 8 x^{9} + 54 x^{8} + 72 x^{7} - 64 x^{6} - 216 x^{5} - 183 x^{4} + 104 x^{3} + 396 x^{2} + 336 x + 98 \)

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:  \(18260173718028288=2^{35}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.65$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{7} a^{8} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{119} a^{11} + \frac{1}{119} a^{10} + \frac{6}{119} a^{9} - \frac{2}{119} a^{8} + \frac{1}{119} a^{7} + \frac{5}{119} a^{6} + \frac{9}{119} a^{5} + \frac{31}{119} a^{4} + \frac{52}{119} a^{3} + \frac{3}{119} a^{2} + \frac{8}{119} a + \frac{3}{17}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6209.56867086 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 12T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.2048.2, 6.2.11943936.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.8.24.7$x^{8} + 8 x^{7} + 12 x^{6} + 10 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.12.12.12$x^{12} + 165 x^{10} - 312 x^{9} - 288 x^{8} - 180 x^{7} - 36 x^{6} - 135 x^{5} - 243 x^{4} + 54 x^{3} + 81 x^{2} + 81 x - 162$$3$$4$$12$12T41$[3/2, 3/2]_{2}^{4}$