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Properties

Label 12.0.13162170384...136.39
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 3^{22}$
Root discriminant $26.71$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3^2$ (as 12T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 0, -324, -72, 837, 324, -432, -144, 81, -12, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 12*x^9 + 81*x^8 - 144*x^7 - 432*x^6 + 324*x^5 + 837*x^4 - 72*x^3 - 324*x^2 + 36)
 
gp: K = bnfinit(x^12 + 18*x^10 - 12*x^9 + 81*x^8 - 144*x^7 - 432*x^6 + 324*x^5 + 837*x^4 - 72*x^3 - 324*x^2 + 36, 1)
 

Normalized defining polynomial

\( x^{12} + 18 x^{10} - 12 x^{9} + 81 x^{8} - 144 x^{7} - 432 x^{6} + 324 x^{5} + 837 x^{4} - 72 x^{3} - 324 x^{2} + 36 \)

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:  \(131621703842267136=2^{22}\cdot 3^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{21} a^{9} + \frac{2}{21} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{168} a^{10} - \frac{1}{14} a^{8} + \frac{1}{42} a^{7} - \frac{1}{24} a^{6} + \frac{2}{7} a^{5} + \frac{11}{28} a^{4} + \frac{5}{14} a^{3} - \frac{13}{56} a^{2} + \frac{1}{7} a - \frac{1}{4}$, $\frac{1}{759834768} a^{11} - \frac{57121}{47489673} a^{10} + \frac{669105}{63319564} a^{9} - \frac{3835861}{27136956} a^{8} - \frac{21954887}{759834768} a^{7} + \frac{12166793}{94979346} a^{6} - \frac{2597059}{11512648} a^{5} + \frac{6420551}{63319564} a^{4} + \frac{83612755}{253278256} a^{3} + \frac{15525}{52591} a^{2} + \frac{6933481}{126639128} a - \frac{911957}{15829891}$

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:  \( -\frac{339921}{841456} a^{11} - \frac{7324}{52591} a^{10} - \frac{4645481}{631092} a^{9} + \frac{486063}{210364} a^{8} - \frac{27500889}{841456} a^{7} + \frac{4986785}{105182} a^{6} + \frac{7164585}{38248} a^{5} - \frac{12672117}{210364} a^{4} - \frac{287543161}{841456} a^{3} - \frac{5343840}{52591} a^{2} + \frac{27582669}{420728} a + \frac{1294826}{52591} \) (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:  \( 25074.0665739 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 12T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.0.362797056.6

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.12.22.81$x^{12} + 18 x^{11} - 27 x^{10} - 15 x^{9} - 9 x^{8} - 27 x^{7} + 9 x^{6} + 27 x^{5} - 27 x^{4} - 9 x^{3} + 27 x + 9$$6$$2$$22$$S_3^2$$[2, 5/2]_{2}^{2}$