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Properties

Label 12.0.55142971819...3041.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 71^{6}$
Root discriminant $36.46$
Ramified primes $3, 71$
Class number $42$
Class group $[42]$
Galois group $A_4 \times C_2$ (as 12T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29241, 16929, 34830, 7056, 10584, -72, 1899, -54, 180, -19, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 21*x^10 - 19*x^9 + 180*x^8 - 54*x^7 + 1899*x^6 - 72*x^5 + 10584*x^4 + 7056*x^3 + 34830*x^2 + 16929*x + 29241)
 
gp: K = bnfinit(x^12 - 3*x^11 + 21*x^10 - 19*x^9 + 180*x^8 - 54*x^7 + 1899*x^6 - 72*x^5 + 10584*x^4 + 7056*x^3 + 34830*x^2 + 16929*x + 29241, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 21 x^{10} - 19 x^{9} + 180 x^{8} - 54 x^{7} + 1899 x^{6} - 72 x^{5} + 10584 x^{4} + 7056 x^{3} + 34830 x^{2} + 16929 x + 29241 \)

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:  \(5514297181968073041=3^{16}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.46$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{3} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{36} a^{9} - \frac{1}{12} a^{8} - \frac{1}{36} a^{6} + \frac{1}{12} a^{5} + \frac{5}{12} a^{4} - \frac{5}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{684} a^{10} + \frac{1}{76} a^{9} - \frac{7}{114} a^{8} + \frac{47}{684} a^{7} + \frac{1}{228} a^{6} + \frac{13}{228} a^{5} - \frac{3}{76} a^{4} - \frac{47}{114} a^{3} - \frac{9}{19} a^{2} - \frac{9}{76} a$, $\frac{1}{2573265833036532} a^{11} + \frac{79810969615}{285918425892948} a^{10} + \frac{128063097587}{47653070982158} a^{9} + \frac{161996428977911}{2573265833036532} a^{8} - \frac{41946560143423}{857755277678844} a^{7} - \frac{54387048443423}{857755277678844} a^{6} - \frac{6064726196945}{95306141964316} a^{5} - \frac{33872880848233}{142959212946474} a^{4} - \frac{10831891068571}{142959212946474} a^{3} + \frac{38875342146499}{285918425892948} a^{2} - \frac{3191464371175}{23826535491079} a + \frac{298563280010}{1254028183741}$

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

Class group and class number

$C_{42}$, which has order $42$

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:  \( -1 \) (order $2$)
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:  \( 3039.60843665 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4$ (as 12T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 8 conjugacy class representatives for $A_4 \times C_2$
Character table for $A_4 \times C_2$

Intermediate fields

\(\Q(\sqrt{-71}) \), \(\Q(\zeta_{9})^+\), 6.0.2348254071.3, 6.0.465831.1, 6.6.33074001.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 8 sibling: data not computed
Degree 12 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$