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Properties

Label 12.0.13718968384000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 5^{6}\cdot 11^{8}$
Root discriminant $22.12$
Ramified primes $2, 5, 11$
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![64, -192, 288, -176, -12, 104, -16, 52, -3, -22, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 18*x^10 - 22*x^9 - 3*x^8 + 52*x^7 - 16*x^6 + 104*x^5 - 12*x^4 - 176*x^3 + 288*x^2 - 192*x + 64)
 
gp: K = bnfinit(x^12 - 6*x^11 + 18*x^10 - 22*x^9 - 3*x^8 + 52*x^7 - 16*x^6 + 104*x^5 - 12*x^4 - 176*x^3 + 288*x^2 - 192*x + 64, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 18 x^{10} - 22 x^{9} - 3 x^{8} + 52 x^{7} - 16 x^{6} + 104 x^{5} - 12 x^{4} - 176 x^{3} + 288 x^{2} - 192 x + 64 \)

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:  \(13718968384000000=2^{12}\cdot 5^{6}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.12$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{8} - \frac{1}{4} a^{5} - \frac{7}{40} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{2}{5}$, $\frac{1}{80} a^{9} + \frac{1}{8} a^{6} - \frac{7}{80} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a$, $\frac{1}{2080} a^{10} - \frac{1}{208} a^{9} - \frac{11}{1040} a^{8} - \frac{7}{208} a^{7} - \frac{147}{2080} a^{6} + \frac{19}{130} a^{4} + \frac{3}{26} a^{3} - \frac{161}{520} a^{2} - \frac{1}{13} a - \frac{37}{130}$, $\frac{1}{128960} a^{11} + \frac{1}{64480} a^{10} - \frac{9}{12896} a^{9} + \frac{249}{64480} a^{8} - \frac{6187}{128960} a^{7} + \frac{5929}{32240} a^{6} + \frac{725}{6448} a^{5} - \frac{1607}{16120} a^{4} + \frac{273}{2480} a^{3} + \frac{1197}{8060} a^{2} - \frac{307}{1612} a - \frac{748}{2015}$

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:  \( \frac{1743}{128960} a^{11} - \frac{2461}{32240} a^{10} + \frac{14007}{64480} a^{9} - \frac{14191}{64480} a^{8} - \frac{1357}{9920} a^{7} + \frac{47449}{64480} a^{6} - \frac{2277}{32240} a^{5} + \frac{2806}{2015} a^{4} + \frac{18147}{32240} a^{3} - \frac{29463}{16120} a^{2} + \frac{28129}{8060} a - \frac{5791}{4030} \) (order $4$)
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:  \( 3877.32671982 \)
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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), 3.1.2420.1 x3, \(\Q(i, \sqrt{5})\), 6.0.117128000.2, 6.2.29282000.1 x3, 6.0.23425600.1 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$