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Properties

Label 12.0.10437554603...3296.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{6}\cdot 7^{8}\cdot 23^{6}$
Root discriminant $121.59$
Ramified primes $2, 3, 7, 23$
Class number $116064$ (GRH)
Class group $[2, 12, 4836]$ (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![2404591561, -241593532, 365351184, -30808716, 23739599, -1633236, 847622, -45212, 17604, -656, 202, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 202*x^10 - 656*x^9 + 17604*x^8 - 45212*x^7 + 847622*x^6 - 1633236*x^5 + 23739599*x^4 - 30808716*x^3 + 365351184*x^2 - 241593532*x + 2404591561)
 
gp: K = bnfinit(x^12 - 4*x^11 + 202*x^10 - 656*x^9 + 17604*x^8 - 45212*x^7 + 847622*x^6 - 1633236*x^5 + 23739599*x^4 - 30808716*x^3 + 365351184*x^2 - 241593532*x + 2404591561, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 202 x^{10} - 656 x^{9} + 17604 x^{8} - 45212 x^{7} + 847622 x^{6} - 1633236 x^{5} + 23739599 x^{4} - 30808716 x^{3} + 365351184 x^{2} - 241593532 x + 2404591561 \)

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:  \(10437554603200547490103296=2^{24}\cdot 3^{6}\cdot 7^{8}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.59$
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:  $2, 3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3864=2^{3}\cdot 3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{3864}(1,·)$, $\chi_{3864}(3035,·)$, $\chi_{3864}(1381,·)$, $\chi_{3864}(2209,·)$, $\chi_{3864}(2759,·)$, $\chi_{3864}(1933,·)$, $\chi_{3864}(1103,·)$, $\chi_{3864}(3313,·)$, $\chi_{3864}(275,·)$, $\chi_{3864}(277,·)$, $\chi_{3864}(827,·)$, $\chi_{3864}(2207,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{35} a^{9} - \frac{3}{35} a^{8} - \frac{9}{35} a^{7} - \frac{1}{5} a^{6} + \frac{12}{35} a^{5} + \frac{3}{7} a^{4} + \frac{3}{35} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{35}$, $\frac{1}{710867255} a^{10} - \frac{820142}{710867255} a^{9} + \frac{104013118}{710867255} a^{8} - \frac{19153271}{710867255} a^{7} + \frac{14565419}{142173451} a^{6} + \frac{332150067}{710867255} a^{5} + \frac{228328133}{710867255} a^{4} - \frac{100775987}{710867255} a^{3} - \frac{55256235}{142173451} a^{2} - \frac{313839069}{710867255} a - \frac{223063594}{710867255}$, $\frac{1}{132091472370054137255} a^{11} - \frac{81649434569}{132091472370054137255} a^{10} - \frac{407309875483016344}{132091472370054137255} a^{9} + \frac{3268922730165606981}{132091472370054137255} a^{8} + \frac{26340521032858964966}{132091472370054137255} a^{7} - \frac{772630409895961406}{132091472370054137255} a^{6} + \frac{29317745496973696557}{132091472370054137255} a^{5} + \frac{3442115526115080761}{18870210338579162465} a^{4} - \frac{54927752975841367294}{132091472370054137255} a^{3} - \frac{42550968470139047479}{132091472370054137255} a^{2} - \frac{398305602975312146}{1860443272817663905} a - \frac{33523828610401638933}{132091472370054137255}$

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

Class group and class number

$C_{2}\times C_{12}\times C_{4836}$, which has order $116064$ (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

\(\Q(\sqrt{-138}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-69})\), 6.0.403840055808.5, 6.0.50480006976.3, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.24.318$x^{12} + 60 x^{11} + 14 x^{10} + 36 x^{9} - 34 x^{8} - 32 x^{7} - 48 x^{6} - 32 x^{5} + 36 x^{4} - 16 x^{3} - 40 x^{2} - 48 x + 56$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$