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Properties

Label 12.0.2779905883635712.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 13^{9}$
Root discriminant $19.36$
Ramified primes $2, 13$
Class number $3$
Class group $[3]$
Galois group $D_4 \times C_3$ (as 12T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -206, 384, -228, 266, -204, 172, -50, -17, 30, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125)
 
gp: K = bnfinit(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 6 x^{10} + 30 x^{9} - 17 x^{8} - 50 x^{7} + 172 x^{6} - 204 x^{5} + 266 x^{4} - 228 x^{3} + 384 x^{2} - 206 x + 125 \)

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:  \(2779905883635712=2^{18}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.36$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{265} a^{9} - \frac{4}{53} a^{8} + \frac{128}{265} a^{7} - \frac{112}{265} a^{6} + \frac{91}{265} a^{5} + \frac{10}{53} a^{4} + \frac{62}{265} a^{3} - \frac{93}{265} a^{2} - \frac{122}{265} a + \frac{3}{53}$, $\frac{1}{265} a^{10} - \frac{7}{265} a^{8} + \frac{63}{265} a^{7} - \frac{29}{265} a^{6} + \frac{3}{53} a^{5} + \frac{2}{265} a^{4} + \frac{87}{265} a^{3} - \frac{127}{265} a^{2} - \frac{8}{53} a + \frac{7}{53}$, $\frac{1}{1783504325} a^{11} - \frac{468629}{356700865} a^{10} - \frac{2448941}{1783504325} a^{9} + \frac{234624333}{1783504325} a^{8} - \frac{639305346}{1783504325} a^{7} + \frac{573936963}{1783504325} a^{6} + \frac{633973138}{1783504325} a^{5} + \frac{137766927}{1783504325} a^{4} + \frac{112233513}{356700865} a^{3} + \frac{168916367}{1783504325} a^{2} + \frac{709522173}{1783504325} a + \frac{17909263}{71340173}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{134877}{16362425} a^{11} + \frac{40892}{3272485} a^{10} + \frac{937757}{16362425} a^{9} - \frac{3610431}{16362425} a^{8} + \frac{515152}{16362425} a^{7} + \frac{7421294}{16362425} a^{6} - \frac{20663451}{16362425} a^{5} + \frac{19764911}{16362425} a^{4} - \frac{3928168}{3272485} a^{3} + \frac{10675001}{16362425} a^{2} - \frac{31607796}{16362425} a + \frac{374126}{654497} \) (order $4$)
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:  \( 919.513377747 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_4$ (as 12T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 15 conjugacy class representatives for $D_4 \times C_3$
Character table for $D_4 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.169.1, 4.0.832.1, 6.0.1827904.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.18.63$x^{12} + 6 x^{11} + 8 x^{10} - 52 x^{9} - 10 x^{8} + 24 x^{7} + 8 x^{6} + 64 x^{5} + 28 x^{4} - 40 x^{3} - 16 x^{2} - 16 x + 40$$4$$3$$18$$D_4 \times C_3$$[2, 2]^{6}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$