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Properties

Label 12.0.4244902593608889.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{14}\cdot 31^{6}$
Root discriminant $20.06$
Ramified primes $3, 31$
Class number $3$
Class group $[3]$
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![100, -90, -99, 258, -15, -168, 137, -108, 81, -48, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 21*x^10 - 48*x^9 + 81*x^8 - 108*x^7 + 137*x^6 - 168*x^5 - 15*x^4 + 258*x^3 - 99*x^2 - 90*x + 100)
 
gp: K = bnfinit(x^12 - 6*x^11 + 21*x^10 - 48*x^9 + 81*x^8 - 108*x^7 + 137*x^6 - 168*x^5 - 15*x^4 + 258*x^3 - 99*x^2 - 90*x + 100, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 21 x^{10} - 48 x^{9} + 81 x^{8} - 108 x^{7} + 137 x^{6} - 168 x^{5} - 15 x^{4} + 258 x^{3} - 99 x^{2} - 90 x + 100 \)

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:  \(4244902593608889=3^{14}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.06$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{8} - \frac{5}{12} a^{2}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{6} a^{3} + \frac{1}{8} a^{2} - \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{144} a^{10} - \frac{1}{48} a^{9} - \frac{1}{144} a^{8} - \frac{1}{72} a^{6} - \frac{1}{24} a^{5} - \frac{11}{144} a^{4} - \frac{5}{48} a^{3} - \frac{25}{144} a^{2} - \frac{1}{24} a + \frac{1}{36}$, $\frac{1}{393840} a^{11} - \frac{11}{24615} a^{10} - \frac{2807}{196920} a^{9} + \frac{10757}{393840} a^{8} + \frac{6023}{196920} a^{7} + \frac{4933}{98460} a^{6} + \frac{31627}{393840} a^{5} + \frac{6869}{49230} a^{4} - \frac{18781}{39384} a^{3} + \frac{2303}{393840} a^{2} + \frac{18893}{196920} a - \frac{5293}{19692}$

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:  \( \frac{83}{32820} a^{11} - \frac{997}{131280} a^{10} + \frac{4157}{131280} a^{9} - \frac{8791}{131280} a^{8} + \frac{11291}{65640} a^{7} - \frac{4357}{16410} a^{6} + \frac{15851}{32820} a^{5} - \frac{61291}{131280} a^{4} + \frac{10315}{26256} a^{3} - \frac{140689}{131280} a^{2} - \frac{31699}{65640} a + \frac{5705}{6564} \) (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:  \( 2491.38208894 \)
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

\(\Q(\sqrt{93}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-31}) \), 3.3.837.1 x3, \(\Q(\sqrt{-3}, \sqrt{-31})\), 6.6.65152917.1, 6.0.2101707.2 x3, 6.0.21717639.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.12.14.11$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$$[3/2]_{2}^{2}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$