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Properties

Label 12.0.1222605980803089.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 109^{6}$
Root discriminant $18.08$
Ramified primes $3, 109$
Class number $2$
Class group $[2]$
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![471, -1203, 1678, -1433, 786, -373, 308, -225, 74, 5, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 6*x^10 + 5*x^9 + 74*x^8 - 225*x^7 + 308*x^6 - 373*x^5 + 786*x^4 - 1433*x^3 + 1678*x^2 - 1203*x + 471)
 
gp: K = bnfinit(x^12 - 2*x^11 - 6*x^10 + 5*x^9 + 74*x^8 - 225*x^7 + 308*x^6 - 373*x^5 + 786*x^4 - 1433*x^3 + 1678*x^2 - 1203*x + 471, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 6 x^{10} + 5 x^{9} + 74 x^{8} - 225 x^{7} + 308 x^{6} - 373 x^{5} + 786 x^{4} - 1433 x^{3} + 1678 x^{2} - 1203 x + 471 \)

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:  \(1222605980803089=3^{6}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.08$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{3} a^{4} - \frac{4}{9} a^{2} - \frac{1}{3}$, $\frac{1}{45} a^{9} - \frac{1}{45} a^{8} - \frac{2}{15} a^{7} - \frac{1}{5} a^{5} - \frac{2}{15} a^{4} + \frac{8}{45} a^{3} - \frac{14}{45} a^{2} - \frac{4}{15} a + \frac{4}{15}$, $\frac{1}{135} a^{10} - \frac{7}{135} a^{8} - \frac{2}{45} a^{7} + \frac{7}{45} a^{6} + \frac{2}{9} a^{5} + \frac{32}{135} a^{4} - \frac{2}{45} a^{3} + \frac{4}{135} a^{2} + \frac{1}{3} a - \frac{11}{45}$, $\frac{1}{1574298315} a^{11} - \frac{287728}{1574298315} a^{10} + \frac{2743912}{314859663} a^{9} - \frac{43541117}{1574298315} a^{8} + \frac{11137393}{174922035} a^{7} - \frac{5275319}{58307345} a^{6} + \frac{559624034}{1574298315} a^{5} + \frac{112178596}{1574298315} a^{4} + \frac{579890023}{1574298315} a^{3} + \frac{45240175}{314859663} a^{2} + \frac{139378696}{524766105} a + \frac{240840101}{524766105}$

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:  \( -\frac{561}{66485} a^{11} + \frac{1531}{598365} a^{10} + \frac{38101}{598365} a^{9} + \frac{4231}{66485} a^{8} - \frac{116531}{199455} a^{7} + \frac{167267}{199455} a^{6} - \frac{7705}{13297} a^{5} + \frac{784013}{598365} a^{4} - \frac{2325118}{598365} a^{3} + \frac{920099}{199455} a^{2} - \frac{612898}{199455} a + \frac{90172}{66485} \) (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:  \( 3227.94367202 \)
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

\(\Q(\sqrt{-327}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{109}) \), 3.1.327.1 x3, \(\Q(\sqrt{-3}, \sqrt{109})\), 6.0.34965783.1, 6.0.320787.2 x3, 6.2.11655261.1 x3

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

Sibling fields

Degree 6 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$109$109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$