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Properties

Label 12.0.53175536981...6784.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{8}\cdot 13^{6}$
Root discriminant $64.64$
Ramified primes $2, 3, 7, 13$
Class number $2432$ (GRH)
Class group $[4, 4, 152]$ (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![4495807, -476098, 1468591, -125512, 209715, -16158, 17923, -1282, 999, -64, 39, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 39*x^10 - 64*x^9 + 999*x^8 - 1282*x^7 + 17923*x^6 - 16158*x^5 + 209715*x^4 - 125512*x^3 + 1468591*x^2 - 476098*x + 4495807)
 
gp: K = bnfinit(x^12 - 2*x^11 + 39*x^10 - 64*x^9 + 999*x^8 - 1282*x^7 + 17923*x^6 - 16158*x^5 + 209715*x^4 - 125512*x^3 + 1468591*x^2 - 476098*x + 4495807, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 39 x^{10} - 64 x^{9} + 999 x^{8} - 1282 x^{7} + 17923 x^{6} - 16158 x^{5} + 209715 x^{4} - 125512 x^{3} + 1468591 x^{2} - 476098 x + 4495807 \)

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:  \(5317553698152529526784=2^{18}\cdot 3^{6}\cdot 7^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.64$
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$$\chi_{2184}(1,·)$, $\chi_{2184}(389,·)$, $\chi_{2184}(1793,·)$, $\chi_{2184}(1481,·)$, $\chi_{2184}(781,·)$, $\chi_{2184}(1325,·)$, $\chi_{2184}(625,·)$, $\chi_{2184}(1717,·)$, $\chi_{2184}(233,·)$, $\chi_{2184}(701,·)$, $\chi_{2184}(1873,·)$, $\chi_{2184}(1093,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{47} a^{9} + \frac{22}{47} a^{8} + \frac{17}{47} a^{7} - \frac{12}{47} a^{6} + \frac{18}{47} a^{5} + \frac{8}{47} a^{4} + \frac{15}{47} a^{3} - \frac{2}{47} a^{2} + \frac{21}{47} a - \frac{12}{47}$, $\frac{1}{159118171} a^{10} - \frac{838384}{159118171} a^{9} - \frac{1624702}{3385493} a^{8} + \frac{47247667}{159118171} a^{7} - \frac{38949911}{159118171} a^{6} + \frac{25907364}{159118171} a^{5} + \frac{32041024}{159118171} a^{4} + \frac{16510187}{159118171} a^{3} + \frac{70814021}{159118171} a^{2} - \frac{59110452}{159118171} a - \frac{5415147}{159118171}$, $\frac{1}{2545831117762807891} a^{11} + \frac{3786296789}{2545831117762807891} a^{10} + \frac{134360751965854}{54166619526868253} a^{9} - \frac{592404561714159196}{2545831117762807891} a^{8} + \frac{680251002699530249}{2545831117762807891} a^{7} - \frac{782455742633123587}{2545831117762807891} a^{6} + \frac{773038449101347817}{2545831117762807891} a^{5} - \frac{1010232113507725061}{2545831117762807891} a^{4} + \frac{1187984883566733720}{2545831117762807891} a^{3} - \frac{2878070996528589}{35856776306518421} a^{2} - \frac{52708691537045098}{2545831117762807891} a + \frac{878956768105222991}{2545831117762807891}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{152}$, which has order $2432$ (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:  \( 279.1500271937239 \) (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

\(\Q(\sqrt{-78}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-39})\), 6.0.72921558528.9, 6.0.142424919.1, 6.6.1229312.1

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 ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$