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Properties

Label 12.0.24257976983...4049.4
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{10}\cdot 17^{6}$
Root discriminant $60.54$
Ramified primes $3, 13, 17$
Class number $3024$ (GRH)
Class group $[2, 6, 252]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9579751, -1963242, 3299030, -605817, 510913, -80635, 45212, -5806, 2405, -228, 73, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 73*x^10 - 228*x^9 + 2405*x^8 - 5806*x^7 + 45212*x^6 - 80635*x^5 + 510913*x^4 - 605817*x^3 + 3299030*x^2 - 1963242*x + 9579751)
 
gp: K = bnfinit(x^12 - 4*x^11 + 73*x^10 - 228*x^9 + 2405*x^8 - 5806*x^7 + 45212*x^6 - 80635*x^5 + 510913*x^4 - 605817*x^3 + 3299030*x^2 - 1963242*x + 9579751, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 73 x^{10} - 228 x^{9} + 2405 x^{8} - 5806 x^{7} + 45212 x^{6} - 80635 x^{5} + 510913 x^{4} - 605817 x^{3} + 3299030 x^{2} - 1963242 x + 9579751 \)

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:  \(2425797698386816634049=3^{6}\cdot 13^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.54$
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:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(663=3\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{663}(256,·)$, $\chi_{663}(1,·)$, $\chi_{663}(101,·)$, $\chi_{663}(103,·)$, $\chi_{663}(458,·)$, $\chi_{663}(205,·)$, $\chi_{663}(560,·)$, $\chi_{663}(562,·)$, $\chi_{663}(662,·)$, $\chi_{663}(407,·)$, $\chi_{663}(152,·)$, $\chi_{663}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{26835322188730935551207645} a^{11} + \frac{2225620243453693343515303}{26835322188730935551207645} a^{10} - \frac{163126056490337095402910}{5367064437746187110241529} a^{9} - \frac{1918006158699983850991543}{5367064437746187110241529} a^{8} + \frac{2035654571263302189447069}{5367064437746187110241529} a^{7} + \frac{5497919299932771674458214}{26835322188730935551207645} a^{6} + \frac{294742075347558425242335}{5367064437746187110241529} a^{5} + \frac{17438507606363394841929}{26835322188730935551207645} a^{4} + \frac{12549925584434415935881476}{26835322188730935551207645} a^{3} - \frac{8737392611979442969773126}{26835322188730935551207645} a^{2} + \frac{9637336540443406964197614}{26835322188730935551207645} a + \frac{1983690454681326920339046}{5367064437746187110241529}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{252}$, which has order $3024$ (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:  \( 120.78403136265631 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-663}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-51}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-51})\), 6.0.49252387743.2, \(\Q(\zeta_{13})^+\), 6.0.3788645211.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.6.3.1$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$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$