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Properties

Label 12.0.26056013120...2416.8
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 13^{6}$
Root discriminant $89.40$
Ramified primes $2, 3, 7, 13$
Class number $12096$ (GRH)
Class group $[2, 2, 6, 6, 84]$ (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![13749723, -2456298, 2888181, -902538, 550392, -125154, 52261, -9438, 2868, -350, 81, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 81*x^10 - 350*x^9 + 2868*x^8 - 9438*x^7 + 52261*x^6 - 125154*x^5 + 550392*x^4 - 902538*x^3 + 2888181*x^2 - 2456298*x + 13749723)
 
gp: K = bnfinit(x^12 - 6*x^11 + 81*x^10 - 350*x^9 + 2868*x^8 - 9438*x^7 + 52261*x^6 - 125154*x^5 + 550392*x^4 - 902538*x^3 + 2888181*x^2 - 2456298*x + 13749723, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 81 x^{10} - 350 x^{9} + 2868 x^{8} - 9438 x^{7} + 52261 x^{6} - 125154 x^{5} + 550392 x^{4} - 902538 x^{3} + 2888181 x^{2} - 2456298 x + 13749723 \)

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:  \(260560131209473946812416=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.40$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2184=2^{3}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2184}(1,·)$, $\chi_{2184}(1091,·)$, $\chi_{2184}(131,·)$, $\chi_{2184}(961,·)$, $\chi_{2184}(1067,·)$, $\chi_{2184}(337,·)$, $\chi_{2184}(467,·)$, $\chi_{2184}(755,·)$, $\chi_{2184}(625,·)$, $\chi_{2184}(25,·)$, $\chi_{2184}(1403,·)$, $\chi_{2184}(1873,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{5} + \frac{1}{36} a^{4} - \frac{1}{6} a^{3} + \frac{1}{4}$, $\frac{1}{36} a^{9} + \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{1}{12} a^{5} - \frac{1}{18} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{4} a$, $\frac{1}{40771170756} a^{10} - \frac{5}{40771170756} a^{9} - \frac{34892885}{4530130084} a^{8} + \frac{209357315}{6795195126} a^{7} + \frac{2199349373}{40771170756} a^{6} + \frac{865279499}{13590390252} a^{5} - \frac{284751367}{40771170756} a^{4} + \frac{3393209479}{6795195126} a^{3} - \frac{1318699169}{4530130084} a^{2} - \frac{1549278753}{4530130084} a + \frac{94673577}{4530130084}$, $\frac{1}{9466617366664884} a^{11} + \frac{116089}{9466617366664884} a^{10} - \frac{3541841661389}{1051846374073876} a^{9} + \frac{35662245445159}{3155539122221628} a^{8} - \frac{558604525012009}{9466617366664884} a^{7} - \frac{105068103740551}{3155539122221628} a^{6} - \frac{414417684072985}{9466617366664884} a^{5} - \frac{142328535258637}{3155539122221628} a^{4} - \frac{4484303861471}{10843777052308} a^{3} - \frac{5234109932475}{10843777052308} a^{2} + \frac{371330379768285}{1051846374073876} a + \frac{387785736517459}{1051846374073876}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{84}$, which has order $12096$ (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:  \( 562.7753300008496 \) (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

\(\Q(\sqrt{-546}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{13}, \sqrt{-42})\), 6.0.510450909696.3, 6.0.232339968.1, 6.6.5274997.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$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$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$