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Properties

Label 12.0.3217569633140625.3
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $19.60$
Ramified primes $3, 5, 7$
Class number $2$
Class group $[2]$
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -1024, -768, 448, 80, -132, 13, -33, 5, 7, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 3*x^10 + 7*x^9 + 5*x^8 - 33*x^7 + 13*x^6 - 132*x^5 + 80*x^4 + 448*x^3 - 768*x^2 - 1024*x + 4096)
 
gp: K = bnfinit(x^12 - x^11 - 3*x^10 + 7*x^9 + 5*x^8 - 33*x^7 + 13*x^6 - 132*x^5 + 80*x^4 + 448*x^3 - 768*x^2 - 1024*x + 4096, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 3 x^{10} + 7 x^{9} + 5 x^{8} - 33 x^{7} + 13 x^{6} - 132 x^{5} + 80 x^{4} + 448 x^{3} - 768 x^{2} - 1024 x + 4096 \)

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:  \(3217569633140625=3^{6}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.60$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(1,·)$, $\chi_{105}(104,·)$, $\chi_{105}(44,·)$, $\chi_{105}(74,·)$, $\chi_{105}(76,·)$, $\chi_{105}(46,·)$, $\chi_{105}(29,·)$, $\chi_{105}(16,·)$, $\chi_{105}(89,·)$, $\chi_{105}(59,·)$, $\chi_{105}(61,·)$, $\chi_{105}(31,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{52} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{6}{13}$, $\frac{1}{208} a^{8} - \frac{1}{208} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{7}{16} a^{4} - \frac{5}{16} a^{3} + \frac{1}{16} a^{2} + \frac{19}{52} a + \frac{5}{13}$, $\frac{1}{832} a^{9} - \frac{1}{832} a^{8} - \frac{3}{832} a^{7} - \frac{29}{64} a^{6} + \frac{25}{64} a^{5} + \frac{27}{64} a^{4} + \frac{1}{64} a^{3} - \frac{33}{208} a^{2} + \frac{5}{52} a - \frac{6}{13}$, $\frac{1}{3328} a^{10} - \frac{1}{3328} a^{9} - \frac{3}{3328} a^{8} + \frac{7}{3328} a^{7} - \frac{39}{256} a^{6} - \frac{101}{256} a^{5} + \frac{1}{256} a^{4} - \frac{33}{832} a^{3} + \frac{5}{208} a^{2} + \frac{7}{52} a - \frac{3}{13}$, $\frac{1}{13312} a^{11} - \frac{1}{13312} a^{10} - \frac{3}{13312} a^{9} + \frac{7}{13312} a^{8} + \frac{5}{13312} a^{7} + \frac{155}{1024} a^{6} + \frac{1}{1024} a^{5} - \frac{33}{3328} a^{4} + \frac{5}{832} a^{3} + \frac{7}{208} a^{2} - \frac{3}{52} a - \frac{1}{13}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{3}{832} a^{10} + \frac{305}{832} a^{3} \) (order $14$)
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:  \( 1378.58167933 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\zeta_{7})\), 6.6.56723625.1, 6.0.8103375.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$