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Properties

Label 12.0.58498535041007616.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{20}$
Root discriminant $24.96$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group 12T183

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 72, -104, 75, 0, 8, -24, 27, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^9 + 27*x^8 - 24*x^7 + 8*x^6 + 75*x^4 - 104*x^3 + 72*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^12 - 4*x^9 + 27*x^8 - 24*x^7 + 8*x^6 + 75*x^4 - 104*x^3 + 72*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{9} + 27 x^{8} - 24 x^{7} + 8 x^{6} + 75 x^{4} - 104 x^{3} + 72 x^{2} - 12 x + 1 \)

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:  \(58498535041007616=2^{24}\cdot 3^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.96$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{4376} a^{11} + \frac{31}{547} a^{10} - \frac{307}{4376} a^{9} + \frac{55}{547} a^{8} + \frac{147}{2188} a^{7} + \frac{171}{1094} a^{6} + \frac{35}{2188} a^{5} - \frac{18}{547} a^{4} - \frac{1723}{4376} a^{3} - \frac{187}{1094} a^{2} + \frac{1}{4376} a - \frac{244}{547}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{477}{2188} a^{11} - \frac{36}{547} a^{10} - \frac{157}{2188} a^{9} + \frac{1809}{2188} a^{8} - \frac{3060}{547} a^{7} + \frac{2124}{547} a^{6} - \frac{963}{547} a^{5} - \frac{117}{1094} a^{4} - \frac{34731}{2188} a^{3} + \frac{10431}{547} a^{2} - \frac{30015}{2188} a + \frac{2845}{2188} \) (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:  \( 5343.60315738 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T183:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for S_6(12)
Character table for S_6(12)

Intermediate fields

\(\Q(\sqrt{-1}) \), 6.4.60466176.3

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 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 15 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.12.24.432$x^{12} - 2 x^{10} + 4 x^{8} - 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 2$$12$$1$$24$$C_2 \times S_4$$[2, 8/3, 8/3]_{3}^{2}$
$3$3.12.20.30$x^{12} - 24 x^{11} + 15 x^{9} + 27 x^{8} + 18 x^{7} - 9 x^{6} - 18 x^{5} - 36 x^{3} - 27 x^{2} - 27 x + 36$$6$$2$$20$12T34$[9/4, 9/4]_{4}^{2}$