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Properties

Label 12.0.31236051804...0000.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 11^{6}$
Root discriminant $90.76$
Ramified primes $2, 3, 5, 11$
Class number $24696$ (GRH)
Class group $[7, 7, 504]$ (GRH)
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![30496904, -6757728, 7474092, -1480824, 862788, -152892, 61213, -9150, 2751, -320, 75, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 75*x^10 - 320*x^9 + 2751*x^8 - 9150*x^7 + 61213*x^6 - 152892*x^5 + 862788*x^4 - 1480824*x^3 + 7474092*x^2 - 6757728*x + 30496904)
 
gp: K = bnfinit(x^12 - 6*x^11 + 75*x^10 - 320*x^9 + 2751*x^8 - 9150*x^7 + 61213*x^6 - 152892*x^5 + 862788*x^4 - 1480824*x^3 + 7474092*x^2 - 6757728*x + 30496904, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 75 x^{10} - 320 x^{9} + 2751 x^{8} - 9150 x^{7} + 61213 x^{6} - 152892 x^{5} + 862788 x^{4} - 1480824 x^{3} + 7474092 x^{2} - 6757728 x + 30496904 \)

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:  \(312360518047666176000000=2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.76$
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 Galois and abelian over $\Q$.
Conductor:  \(3960=2^{3}\cdot 3^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3960}(769,·)$, $\chi_{3960}(3301,·)$, $\chi_{3960}(1,·)$, $\chi_{3960}(1321,·)$, $\chi_{3960}(109,·)$, $\chi_{3960}(2749,·)$, $\chi_{3960}(3409,·)$, $\chi_{3960}(661,·)$, $\chi_{3960}(2641,·)$, $\chi_{3960}(2089,·)$, $\chi_{3960}(1981,·)$, $\chi_{3960}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{68} a^{8} - \frac{1}{17} a^{7} + \frac{3}{34} a^{6} - \frac{1}{17} a^{5} + \frac{9}{68} a^{4} - \frac{4}{17} a^{3} + \frac{6}{17} a^{2} - \frac{4}{17} a + \frac{5}{17}$, $\frac{1}{7548} a^{9} + \frac{1}{148} a^{8} + \frac{55}{1258} a^{7} - \frac{263}{1258} a^{6} + \frac{315}{2516} a^{5} - \frac{475}{2516} a^{4} - \frac{299}{1887} a^{3} - \frac{84}{629} a^{2} + \frac{189}{629} a + \frac{479}{1887}$, $\frac{1}{14145487908} a^{10} - \frac{5}{14145487908} a^{9} - \frac{2112825}{2357581318} a^{8} + \frac{8451305}{2357581318} a^{7} + \frac{674527709}{4715162636} a^{6} + \frac{274839049}{4715162636} a^{5} - \frac{424547024}{3536371977} a^{4} + \frac{1699125682}{3536371977} a^{3} - \frac{1072803475}{2357581318} a^{2} - \frac{386906233}{3536371977} a + \frac{74269028}{3536371977}$, $\frac{1}{43808576051076} a^{11} + \frac{1543}{43808576051076} a^{10} + \frac{945925767}{14602858683692} a^{9} + \frac{5680815549}{858991687276} a^{8} + \frac{437744827307}{14602858683692} a^{7} - \frac{72654118217}{858991687276} a^{6} - \frac{9305119245893}{43808576051076} a^{5} + \frac{7069667677141}{43808576051076} a^{4} - \frac{898482262429}{7301429341846} a^{3} - \frac{3425055643750}{10952144012769} a^{2} - \frac{2456818065340}{10952144012769} a - \frac{1689951079965}{3650714670923}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{504}$, which has order $24696$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 481.70037561485367 \) (assuming GRH)
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

\(\Q(\sqrt{-110}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-55})\), 6.0.558892224000.14, 6.6.3359232.1, 6.0.1091586375.3

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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$