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Properties

Label 12.0.61464734415...4213.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{11}\cdot 19^{6}$
Root discriminant $79.26$
Ramified primes $3, 13, 19$
Class number $14440$ (GRH)
Class group $[2, 38, 190]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154258651, -56374683, 56374683, -7432699, 7432699, -440987, 440987, -12923, 12923, -183, 183, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 183*x^10 - 183*x^9 + 12923*x^8 - 12923*x^7 + 440987*x^6 - 440987*x^5 + 7432699*x^4 - 7432699*x^3 + 56374683*x^2 - 56374683*x + 154258651)
 
gp: K = bnfinit(x^12 - x^11 + 183*x^10 - 183*x^9 + 12923*x^8 - 12923*x^7 + 440987*x^6 - 440987*x^5 + 7432699*x^4 - 7432699*x^3 + 56374683*x^2 - 56374683*x + 154258651, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 183 x^{10} - 183 x^{9} + 12923 x^{8} - 12923 x^{7} + 440987 x^{6} - 440987 x^{5} + 7432699 x^{4} - 7432699 x^{3} + 56374683 x^{2} - 56374683 x + 154258651 \)

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:  \(61464734415837024654213=3^{6}\cdot 13^{11}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.26$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(741=3\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{741}(512,·)$, $\chi_{741}(1,·)$, $\chi_{741}(227,·)$, $\chi_{741}(683,·)$, $\chi_{741}(172,·)$, $\chi_{741}(685,·)$, $\chi_{741}(398,·)$, $\chi_{741}(400,·)$, $\chi_{741}(626,·)$, $\chi_{741}(628,·)$, $\chi_{741}(571,·)$, $\chi_{741}(284,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{21752431} a^{7} + \frac{539589}{21752431} a^{6} + \frac{98}{21752431} a^{5} + \frac{1820614}{21752431} a^{4} + \frac{2744}{21752431} a^{3} - \frac{5271968}{21752431} a^{2} + \frac{19208}{21752431} a + \frac{2933816}{21752431}$, $\frac{1}{21752431} a^{8} + \frac{112}{21752431} a^{6} - \frac{7554246}{21752431} a^{5} + \frac{3920}{21752431} a^{4} - \frac{6738876}{21752431} a^{3} + \frac{43904}{21752431} a^{2} - \frac{7334540}{21752431} a + \frac{76832}{21752431}$, $\frac{1}{21752431} a^{9} - \frac{2730921}{21752431} a^{6} - \frac{7056}{21752431} a^{5} + \frac{6876666}{21752431} a^{4} - \frac{263424}{21752431} a^{3} - \frac{4189761}{21752431} a^{2} - \frac{2074464}{21752431} a - \frac{2300927}{21752431}$, $\frac{1}{21752431} a^{10} - \frac{8820}{21752431} a^{6} - \frac{8274679}{21752431} a^{5} - \frac{411600}{21752431} a^{4} + \frac{6621199}{21752431} a^{3} - \frac{5186160}{21752431} a^{2} + \frac{8118500}{21752431} a - \frac{9680832}{21752431}$, $\frac{1}{21752431} a^{11} + \frac{8870343}{21752431} a^{6} + \frac{452760}{21752431} a^{5} - \frac{10609830}{21752431} a^{4} - \frac{2736511}{21752431} a^{3} - \frac{5694213}{21752431} a^{2} + \frac{7466711}{21752431} a - \frac{9135770}{21752431}$

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

Class group and class number

$C_{2}\times C_{38}\times C_{190}$, which has order $14440$ (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:  $5$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 120.784031363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.7138053.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$19$19.12.6.2$x^{12} - 2476099 x^{2} + 141137643$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$