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Properties

Label 12.0.93597656065...56.116
Degree $12$
Signature $[0, 6]$
Discriminant $2^{28}\cdot 3^{20}$
Root discriminant $31.45$
Ramified primes $2, 3$
Class number $3$
Class group $[3]$
Galois group $S_3\wr C_2$ (as 12T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, 432, 972, 876, 513, -324, -122, 60, 9, -8, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 8*x^9 + 9*x^8 + 60*x^7 - 122*x^6 - 324*x^5 + 513*x^4 + 876*x^3 + 972*x^2 + 432*x + 144)
 
gp: K = bnfinit(x^12 - 6*x^10 - 8*x^9 + 9*x^8 + 60*x^7 - 122*x^6 - 324*x^5 + 513*x^4 + 876*x^3 + 972*x^2 + 432*x + 144, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{10} - 8 x^{9} + 9 x^{8} + 60 x^{7} - 122 x^{6} - 324 x^{5} + 513 x^{4} + 876 x^{3} + 972 x^{2} + 432 x + 144 \)

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:  \(935976560656121856=2^{28}\cdot 3^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.45$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{9} - \frac{1}{9} a^{6} - \frac{1}{2} a^{5} + \frac{2}{9} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{2160} a^{10} + \frac{7}{360} a^{9} - \frac{1}{40} a^{8} + \frac{5}{108} a^{7} + \frac{19}{720} a^{6} + \frac{43}{120} a^{5} - \frac{49}{1080} a^{4} + \frac{1}{5} a^{3} + \frac{17}{240} a^{2} + \frac{73}{360} a - \frac{1}{20}$, $\frac{1}{19513440} a^{11} - \frac{2023}{9756720} a^{10} + \frac{619}{650448} a^{9} - \frac{218687}{4878360} a^{8} + \frac{1335337}{19513440} a^{7} + \frac{348013}{3252240} a^{6} - \frac{527765}{1951344} a^{5} - \frac{246337}{609795} a^{4} + \frac{273313}{2168160} a^{3} - \frac{455411}{3252240} a^{2} - \frac{510241}{1626120} a - \frac{3339}{22585}$

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:  \( \frac{7187}{2439180} a^{11} - \frac{99}{90340} a^{10} - \frac{6977}{406530} a^{9} - \frac{22121}{1219590} a^{8} + \frac{7571}{271020} a^{7} + \frac{136273}{813060} a^{6} - \frac{502367}{1219590} a^{5} - \frac{7125}{9034} a^{4} + \frac{157181}{90340} a^{3} + \frac{323285}{162612} a^{2} + \frac{66951}{22585} a + \frac{29492}{22585} \) (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:  \( 30413.7353963 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 12T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.322486272.3

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
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.1$x^{8} + 8 x^{5} + 6 x^{4} + 16 x^{3} + 8 x^{2} + 12$$4$$2$$22$$D_4$$[3, 4]^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$