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Properties

Label 12.0.6643012500000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{12}\cdot 5^{11}$
Root discriminant $20.82$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3\times C_3:S_3.C_2$ (as 12T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![80, 480, 1080, 1040, 225, -270, -135, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 135*x^6 - 270*x^5 + 225*x^4 + 1040*x^3 + 1080*x^2 + 480*x + 80)
 
gp: K = bnfinit(x^12 - 135*x^6 - 270*x^5 + 225*x^4 + 1040*x^3 + 1080*x^2 + 480*x + 80, 1)
 

Normalized defining polynomial

\( x^{12} - 135 x^{6} - 270 x^{5} + 225 x^{4} + 1040 x^{3} + 1080 x^{2} + 480 x + 80 \)

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:  \(6643012500000000=2^{8}\cdot 3^{12}\cdot 5^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.82$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{88} a^{9} + \frac{1}{4} a^{8} - \frac{5}{11} a^{7} + \frac{4}{11} a^{6} + \frac{2}{11} a^{5} - \frac{1}{11} a^{4} - \frac{15}{88} a^{3} - \frac{4}{11} a^{2} - \frac{3}{88} a + \frac{7}{44}$, $\frac{1}{704} a^{10} - \frac{1}{176} a^{9} + \frac{23}{176} a^{8} - \frac{5}{22} a^{7} - \frac{7}{44} a^{6} + \frac{3}{11} a^{5} - \frac{71}{704} a^{4} + \frac{135}{352} a^{3} + \frac{213}{704} a^{2} - \frac{87}{176} a + \frac{41}{176}$, $\frac{1}{5632} a^{11} + \frac{1}{2816} a^{10} + \frac{1}{1408} a^{9} - \frac{127}{704} a^{8} - \frac{127}{352} a^{7} - \frac{79}{176} a^{6} + \frac{1465}{5632} a^{5} - \frac{615}{1408} a^{4} - \frac{2135}{5632} a^{3} + \frac{433}{2816} a^{2} + \frac{447}{1408} a + \frac{17}{64}$

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:  \( -\frac{3483}{512} a^{11} + \frac{13635}{2816} a^{10} - \frac{4881}{1408} a^{9} + \frac{1755}{704} a^{8} - \frac{633}{352} a^{7} + \frac{227}{176} a^{6} + \frac{5167071}{5632} a^{5} + \frac{1666629}{1408} a^{4} - \frac{13349889}{5632} a^{3} - \frac{15166845}{2816} a^{2} - \frac{450765}{128} a - \frac{541399}{704} \) (order $10$)
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:  \( 4454.21449848 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times C_3:S_3.C_2$ (as 12T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 18 conjugacy class representatives for $S_3\times C_3:S_3.C_2$
Character table for $S_3\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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 siblings: data not computed
Degree 27 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.12.12.8$x^{12} + 21 x^{11} + 81 x^{10} - 78 x^{9} - 63 x^{8} - 99 x^{7} + 117 x^{6} - 54 x^{5} - 54 x^{4} + 81 x^{2} - 81 x + 81$$3$$4$$12$12T119$[3/2, 3/2, 3/2]_{2}^{4}$
$5$5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$