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Properties

Label 12.0.96938888000000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 59^{4}$
Root discriminant $26.03$
Ramified primes $2, 5, 59$
Class number $6$
Class group $[6]$
Galois group $C_2.S_3^2$ (as 12T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 105, 453, 793, 312, 295, 135, 52, -6, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 - 6*x^9 + 52*x^8 + 135*x^7 + 295*x^6 + 312*x^5 + 793*x^4 + 453*x^3 + 105*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^12 - x^11 + 4*x^10 - 6*x^9 + 52*x^8 + 135*x^7 + 295*x^6 + 312*x^5 + 793*x^4 + 453*x^3 + 105*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 4 x^{10} - 6 x^{9} + 52 x^{8} + 135 x^{7} + 295 x^{6} + 312 x^{5} + 793 x^{4} + 453 x^{3} + 105 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(96938888000000000=2^{12}\cdot 5^{9}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.03$
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:  $2, 5, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{321502444023049} a^{11} + \frac{129251120035285}{321502444023049} a^{10} + \frac{49857288325140}{321502444023049} a^{9} + \frac{100786122214838}{321502444023049} a^{8} - \frac{17495297935645}{321502444023049} a^{7} - \frac{112140012468928}{321502444023049} a^{6} - \frac{120356323647895}{321502444023049} a^{5} + \frac{152589728109310}{321502444023049} a^{4} + \frac{73977777607426}{321502444023049} a^{3} + \frac{12286645188821}{321502444023049} a^{2} - \frac{5827876344237}{321502444023049} a - \frac{60657169703904}{321502444023049}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( -\frac{16842795484854}{321502444023049} a^{11} + \frac{15337262205912}{321502444023049} a^{10} - \frac{66758598936020}{321502444023049} a^{9} + \frac{96629696904770}{321502444023049} a^{8} - \frac{871230524678968}{321502444023049} a^{7} - \frac{2345106059387355}{321502444023049} a^{6} - \frac{5222350341657100}{321502444023049} a^{5} - \frac{5780384690540454}{321502444023049} a^{4} - \frac{14011059669010484}{321502444023049} a^{3} - \frac{8945486888678330}{321502444023049} a^{2} - \frac{3186654836472886}{321502444023049} a - \frac{76963838245518}{321502444023049} \) (order $10$)
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:  \( 2379.33627988 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.S_3^2$ (as 12T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2.S_3^2$
Character table for $C_2.S_3^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 6.6.27848000.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ R

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.6.4.1$x^{6} + 295 x^{3} + 27848$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$