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Properties

Label 12.0.2742118830047232.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{21}$
Root discriminant $19.34$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group 12T235

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 18, 45, 66, 72, 54, 51, 18, -18, -6, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 15*x^10 - 6*x^9 - 18*x^8 + 18*x^7 + 51*x^6 + 54*x^5 + 72*x^4 + 66*x^3 + 45*x^2 + 18*x + 3)
 
gp: K = bnfinit(x^12 - 6*x^11 + 15*x^10 - 6*x^9 - 18*x^8 + 18*x^7 + 51*x^6 + 54*x^5 + 72*x^4 + 66*x^3 + 45*x^2 + 18*x + 3, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 15 x^{10} - 6 x^{9} - 18 x^{8} + 18 x^{7} + 51 x^{6} + 54 x^{5} + 72 x^{4} + 66 x^{3} + 45 x^{2} + 18 x + 3 \)

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:  \(2742118830047232=2^{18}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.34$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{469160} a^{11} + \frac{2471}{58645} a^{10} + \frac{11761}{234580} a^{9} - \frac{106763}{469160} a^{8} - \frac{17503}{93832} a^{7} - \frac{6233}{117290} a^{6} + \frac{11599}{234580} a^{5} - \frac{179819}{469160} a^{4} - \frac{153129}{469160} a^{3} + \frac{5511}{11729} a^{2} - \frac{16089}{46916} a - \frac{6237}{469160}$

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:  \( -\frac{7437}{12680} a^{11} + \frac{6003}{1585} a^{10} - \frac{66487}{6340} a^{9} + \frac{101631}{12680} a^{8} + \frac{19755}{2536} a^{7} - \frac{47639}{3170} a^{6} - \frac{148713}{6340} a^{5} - \frac{248577}{12680} a^{4} - \frac{401547}{12680} a^{3} - \frac{7668}{317} a^{2} - \frac{18045}{1268} a - \frac{36911}{12680} \) (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:  \( 3799.98084807 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T235:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for [2^5]F_36:2_2{S_3^2,t}
Character table for [2^5]F_36:2_2{S_3^2,t} is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.0.944784.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.18.45$x^{8} + 6 x^{4} + 12 x^{2} + 12$$4$$2$$18$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
$3$3.12.21.74$x^{12} - 12 x^{11} - 3 x^{10} + 6 x^{9} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} + 6 x^{3} + 9 x^{2} + 9 x - 6$$12$$1$$21$12T36$[9/4, 9/4]_{4}^{2}$