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Properties

Label 12.0.64022047012...3616.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 11^{6}\cdot 13^{10}$
Root discriminant $79.53$
Ramified primes $2, 11, 13$
Class number $12636$ (GRH)
Class group $[3, 9, 468]$ (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![181540151, -13964626, 39868309, -2612324, 3847558, -208260, 208872, -8858, 6739, -202, 123, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 123*x^10 - 202*x^9 + 6739*x^8 - 8858*x^7 + 208872*x^6 - 208260*x^5 + 3847558*x^4 - 2612324*x^3 + 39868309*x^2 - 13964626*x + 181540151)
 
gp: K = bnfinit(x^12 - 2*x^11 + 123*x^10 - 202*x^9 + 6739*x^8 - 8858*x^7 + 208872*x^6 - 208260*x^5 + 3847558*x^4 - 2612324*x^3 + 39868309*x^2 - 13964626*x + 181540151, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 123 x^{10} - 202 x^{9} + 6739 x^{8} - 8858 x^{7} + 208872 x^{6} - 208260 x^{5} + 3847558 x^{4} - 2612324 x^{3} + 39868309 x^{2} - 13964626 x + 181540151 \)

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:  \(64022047012554352623616=2^{18}\cdot 11^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.53$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1144=2^{3}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1144}(1,·)$, $\chi_{1144}(373,·)$, $\chi_{1144}(901,·)$, $\chi_{1144}(705,·)$, $\chi_{1144}(285,·)$, $\chi_{1144}(529,·)$, $\chi_{1144}(725,·)$, $\chi_{1144}(441,·)$, $\chi_{1144}(1057,·)$, $\chi_{1144}(881,·)$, $\chi_{1144}(989,·)$, $\chi_{1144}(549,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{297945219436543339124782981805} a^{11} + \frac{24161468684673536709826438437}{297945219436543339124782981805} a^{10} - \frac{53537025620241268715290146519}{297945219436543339124782981805} a^{9} + \frac{31060625821280211095558771977}{297945219436543339124782981805} a^{8} + \frac{78940997916596405335133763287}{297945219436543339124782981805} a^{7} - \frac{23526010985035354430028148268}{59589043887308667824956596361} a^{6} - \frac{143147879002571211639489266178}{297945219436543339124782981805} a^{5} + \frac{79956682730243191568186864693}{297945219436543339124782981805} a^{4} - \frac{25081815462991432853837842391}{59589043887308667824956596361} a^{3} - \frac{24030217239785851622530064749}{297945219436543339124782981805} a^{2} - \frac{85452440233356996838084880237}{297945219436543339124782981805} a + \frac{31359058920903344188337740836}{297945219436543339124782981805}$

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

Class group and class number

$C_{3}\times C_{9}\times C_{468}$, which has order $12636$ (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:  \( 120.78403136265631 \) (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

\(\Q(\sqrt{-286}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-22})\), 6.0.253025783296.4, 6.0.19463521792.6, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$