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Properties

Label 12.0.29218823485...5625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{6}\cdot 13^{6}$
Root discriminant $41.89$
Ramified primes $3, 5, 13$
Class number $152$ (GRH)
Class group $[152]$ (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![94429, -50892, 61566, -6522, 14157, -2436, 3551, -1230, 636, -150, 45, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 45*x^10 - 150*x^9 + 636*x^8 - 1230*x^7 + 3551*x^6 - 2436*x^5 + 14157*x^4 - 6522*x^3 + 61566*x^2 - 50892*x + 94429)
 
gp: K = bnfinit(x^12 - 6*x^11 + 45*x^10 - 150*x^9 + 636*x^8 - 1230*x^7 + 3551*x^6 - 2436*x^5 + 14157*x^4 - 6522*x^3 + 61566*x^2 - 50892*x + 94429, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 45 x^{10} - 150 x^{9} + 636 x^{8} - 1230 x^{7} + 3551 x^{6} - 2436 x^{5} + 14157 x^{4} - 6522 x^{3} + 61566 x^{2} - 50892 x + 94429 \)

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:  \(29218823485775015625=3^{18}\cdot 5^{6}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.89$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(1,·)$, $\chi_{585}(194,·)$, $\chi_{585}(196,·)$, $\chi_{585}(389,·)$, $\chi_{585}(391,·)$, $\chi_{585}(584,·)$, $\chi_{585}(79,·)$, $\chi_{585}(274,·)$, $\chi_{585}(116,·)$, $\chi_{585}(469,·)$, $\chi_{585}(311,·)$, $\chi_{585}(506,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{508} a^{10} - \frac{27}{254} a^{9} + \frac{31}{254} a^{8} + \frac{17}{254} a^{7} + \frac{9}{508} a^{6} - \frac{185}{508} a^{5} + \frac{89}{254} a^{4} - \frac{15}{127} a^{3} + \frac{3}{127} a^{2} + \frac{41}{254} a + \frac{187}{508}$, $\frac{1}{419275470602768363852} a^{11} + \frac{166778231041687543}{419275470602768363852} a^{10} - \frac{29733395241944836333}{419275470602768363852} a^{9} + \frac{8051203461691996503}{104818867650692090963} a^{8} + \frac{43125089231519093545}{419275470602768363852} a^{7} - \frac{4063794527607345847}{419275470602768363852} a^{6} - \frac{28667998004202411333}{209637735301384181926} a^{5} - \frac{38338515936760149998}{104818867650692090963} a^{4} - \frac{1715702181011373243}{7910857935901289884} a^{3} + \frac{35024473248234003528}{104818867650692090963} a^{2} + \frac{44283151733797831645}{104818867650692090963} a + \frac{11330722315319113213}{209637735301384181926}$

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

Class group and class number

$C_{152}$, which has order $152$ (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:  \( 201.000834787 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-39})\), 6.6.820125.1, 6.0.5405443875.3, 6.0.43243551.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$