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Properties

Label 12.0.18694897744...8064.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 53^{6}$
Root discriminant $127.64$
Ramified primes $2, 3, 7, 53$
Class number $152880$ (GRH)
Class group $[2, 14, 5460]$ (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![5718892873, -725385798, 765535945, -81537580, 43884395, -3801644, 1372757, -91826, 24665, -1150, 241, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 241*x^10 - 1150*x^9 + 24665*x^8 - 91826*x^7 + 1372757*x^6 - 3801644*x^5 + 43884395*x^4 - 81537580*x^3 + 765535945*x^2 - 725385798*x + 5718892873)
 
gp: K = bnfinit(x^12 - 6*x^11 + 241*x^10 - 1150*x^9 + 24665*x^8 - 91826*x^7 + 1372757*x^6 - 3801644*x^5 + 43884395*x^4 - 81537580*x^3 + 765535945*x^2 - 725385798*x + 5718892873, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 241 x^{10} - 1150 x^{9} + 24665 x^{8} - 91826 x^{7} + 1372757 x^{6} - 3801644 x^{5} + 43884395 x^{4} - 81537580 x^{3} + 765535945 x^{2} - 725385798 x + 5718892873 \)

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:  \(18694897744381964174168064=2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.64$
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:  $2, 3, 7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4452=2^{2}\cdot 3\cdot 7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{4452}(1,·)$, $\chi_{4452}(4451,·)$, $\chi_{4452}(4135,·)$, $\chi_{4452}(3817,·)$, $\chi_{4452}(3499,·)$, $\chi_{4452}(635,·)$, $\chi_{4452}(2545,·)$, $\chi_{4452}(1907,·)$, $\chi_{4452}(3497,·)$, $\chi_{4452}(953,·)$, $\chi_{4452}(955,·)$, $\chi_{4452}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19177256309149} a^{10} - \frac{5}{19177256309149} a^{9} - \frac{6847698823770}{19177256309149} a^{8} + \frac{8213538985961}{19177256309149} a^{7} - \frac{5905304493295}{19177256309149} a^{6} - \frac{1442844816425}{19177256309149} a^{5} - \frac{9126545899966}{19177256309149} a^{4} + \frac{7866829616931}{19177256309149} a^{3} + \frac{4194764557}{19177256309149} a^{2} + \frac{7237830666011}{19177256309149} a - \frac{1567888851821}{19177256309149}$, $\frac{1}{91352989104081577039} a^{11} + \frac{2381800}{91352989104081577039} a^{10} + \frac{21910046840004618003}{91352989104081577039} a^{9} - \frac{23552108396791711292}{91352989104081577039} a^{8} + \frac{21423523839984436545}{91352989104081577039} a^{7} - \frac{40069146158039320026}{91352989104081577039} a^{6} + \frac{18142604360082556010}{91352989104081577039} a^{5} - \frac{12348977887399898322}{91352989104081577039} a^{4} + \frac{12366032610873151040}{91352989104081577039} a^{3} - \frac{15608381088783395482}{91352989104081577039} a^{2} - \frac{17500902824836427595}{91352989104081577039} a - \frac{31792246490244926676}{91352989104081577039}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{5460}$, which has order $152880$ (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:  \( 246.50546308257188 \) (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

\(\Q(\sqrt{-1113}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-159}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{7}, \sqrt{-159})\), 6.0.4323759676992.3, \(\Q(\zeta_{28})^+\), 6.0.9651249279.4

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$53$53.6.3.2$x^{6} - 2809 x^{2} + 1191016$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.2$x^{6} - 2809 x^{2} + 1191016$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$