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Properties

Label 12.0.144054149089536.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{14}\cdot 7^{6}$
Root discriminant $15.13$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $D_6$ (as 12T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -12, 96, -340, 624, -654, 431, -189, 48, 1, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4)
 
gp: K = bnfinit(x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + x^{9} + 48 x^{8} - 189 x^{7} + 431 x^{6} - 654 x^{5} + 624 x^{4} - 340 x^{3} + 96 x^{2} - 12 x + 4 \)

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:  \(144054149089536=2^{8}\cdot 3^{14}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.13$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{14} a^{9} - \frac{1}{14} a^{8} - \frac{3}{7} a^{7} - \frac{1}{14} a^{6} + \frac{3}{7} a^{5} + \frac{1}{14} a^{4} - \frac{5}{14} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{196} a^{10} - \frac{5}{196} a^{9} - \frac{3}{49} a^{8} + \frac{37}{196} a^{7} - \frac{2}{7} a^{6} + \frac{89}{196} a^{5} + \frac{15}{196} a^{4} - \frac{47}{98} a^{3} - \frac{2}{49} a^{2} + \frac{39}{98} a - \frac{20}{49}$, $\frac{1}{5500348} a^{11} - \frac{169}{1375087} a^{10} - \frac{134375}{5500348} a^{9} - \frac{6575}{289492} a^{8} - \frac{2359467}{5500348} a^{7} - \frac{1483029}{5500348} a^{6} + \frac{628990}{1375087} a^{5} - \frac{109535}{289492} a^{4} + \frac{49195}{1375087} a^{3} - \frac{327}{56126} a^{2} - \frac{610051}{2750174} a - \frac{96907}{1375087}$

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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{559}{10339} a^{11} - \frac{4785}{41356} a^{10} - \frac{901}{5908} a^{9} + \frac{47}{1477} a^{8} + \frac{113135}{41356} a^{7} - \frac{80932}{10339} a^{6} + \frac{580379}{41356} a^{5} - \frac{648191}{41356} a^{4} + \frac{106179}{20678} a^{3} + \frac{52092}{10339} a^{2} - \frac{71077}{20678} a + \frac{6829}{10339} \) (order $6$)
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:  \( 812.119885731 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 12T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 3.3.756.1 x3, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.6.12002256.1, 6.0.1714608.1 x3, 6.0.4000752.1 x3

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

Sibling fields

Degree 6 siblings: 6.0.1714608.1, 6.0.4000752.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$