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Properties

Label 12.12.4297083967...6912.2
Degree $12$
Signature $[12, 0]$
Discriminant $2^{33}\cdot 29^{8}$
Root discriminant $63.50$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3902, 1336, -15456, -4192, 15613, 1456, -5804, -32, 864, -8, -52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 52*x^10 - 8*x^9 + 864*x^8 - 32*x^7 - 5804*x^6 + 1456*x^5 + 15613*x^4 - 4192*x^3 - 15456*x^2 + 1336*x + 3902)
 
gp: K = bnfinit(x^12 - 52*x^10 - 8*x^9 + 864*x^8 - 32*x^7 - 5804*x^6 + 1456*x^5 + 15613*x^4 - 4192*x^3 - 15456*x^2 + 1336*x + 3902, 1)
 

Normalized defining polynomial

\( x^{12} - 52 x^{10} - 8 x^{9} + 864 x^{8} - 32 x^{7} - 5804 x^{6} + 1456 x^{5} + 15613 x^{4} - 4192 x^{3} - 15456 x^{2} + 1336 x + 3902 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4297083967217611046912=2^{33}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.50$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{3088} a^{10} + \frac{19}{1544} a^{9} + \frac{65}{3088} a^{8} + \frac{361}{772} a^{7} + \frac{747}{3088} a^{6} + \frac{239}{1544} a^{5} + \frac{759}{3088} a^{4} - \frac{19}{193} a^{3} - \frac{345}{1544} a^{2} - \frac{163}{772} a - \frac{25}{1544}$, $\frac{1}{744306071792} a^{11} + \frac{27355239}{744306071792} a^{10} - \frac{28950309483}{744306071792} a^{9} - \frac{25461676503}{744306071792} a^{8} - \frac{57679814677}{744306071792} a^{7} + \frac{364066018345}{744306071792} a^{6} + \frac{298762098731}{744306071792} a^{5} - \frac{367789388477}{744306071792} a^{4} - \frac{42141162125}{372153035896} a^{3} + \frac{165351969665}{372153035896} a^{2} + \frac{48953806815}{372153035896} a - \frac{44178109125}{372153035896}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29190718.8092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.6728.1 x3, \(\Q(\zeta_{16})^+\), 6.6.362127872.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$29$29.12.8.1$x^{12} - 87 x^{9} + 2523 x^{6} - 24389 x^{3} + 4851240379$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$