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Properties

Label 12.6.176836992750000.1
Degree $12$
Signature $[6, 3]$
Discriminant $-\,2^{4}\cdot 3^{12}\cdot 5^{6}\cdot 11^{3}$
Root discriminant $15.39$
Ramified primes $2, 3, 5, 11$
Class number $1$
Class group Trivial
Galois group 12T235

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 18, -8, -51, -63, -40, -12, 0, -1, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^10 - x^9 - 12*x^7 - 40*x^6 - 63*x^5 - 51*x^4 - 8*x^3 + 18*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^12 - 3*x^10 - x^9 - 12*x^7 - 40*x^6 - 63*x^5 - 51*x^4 - 8*x^3 + 18*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{10} - x^{9} - 12 x^{7} - 40 x^{6} - 63 x^{5} - 51 x^{4} - 8 x^{3} + 18 x^{2} + 9 x + 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-176836992750000=-\,2^{4}\cdot 3^{12}\cdot 5^{6}\cdot 11^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.39$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{753} a^{11} + \frac{113}{753} a^{10} - \frac{35}{753} a^{9} + \frac{20}{251} a^{8} + \frac{1}{251} a^{7} + \frac{76}{753} a^{6} + \frac{265}{753} a^{5} - \frac{238}{753} a^{4} - \frac{88}{753} a^{3} + \frac{88}{753} a^{2} + \frac{173}{753} a + \frac{77}{251}$

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:  $8$
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:  \( 464.973480083 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T235:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for [2^5]F_36:2_2{S_3^2,t}
Character table for [2^5]F_36:2_2{S_3^2,t} is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.4.1002375.1

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 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$3$3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$