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Properties

Label 12.0.42288257911...6361.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 127^{10}$
Root discriminant $294.34$
Ramified primes $3, 127$
Class number $1290240$ (GRH)
Class group $[4, 4, 48, 1680]$ (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![32005984375, 0, 0, -64516000, 0, 0, 822833, 0, 0, -1778, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 1778*x^9 + 822833*x^6 - 64516000*x^3 + 32005984375)
 
gp: K = bnfinit(x^12 - 1778*x^9 + 822833*x^6 - 64516000*x^3 + 32005984375, 1)
 

Normalized defining polynomial

\( x^{12} - 1778 x^{9} + 822833 x^{6} - 64516000 x^{3} + 32005984375 \)

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:  \(422882579117748274920892556361=3^{18}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $294.34$
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:  $3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1143=3^{2}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{1143}(1,·)$, $\chi_{1143}(869,·)$, $\chi_{1143}(742,·)$, $\chi_{1143}(362,·)$, $\chi_{1143}(781,·)$, $\chi_{1143}(401,·)$, $\chi_{1143}(274,·)$, $\chi_{1143}(1142,·)$, $\chi_{1143}(890,·)$, $\chi_{1143}(997,·)$, $\chi_{1143}(253,·)$, $\chi_{1143}(146,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{254} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{1270} a^{7} + \frac{1}{10} a^{4} - \frac{1}{10} a$, $\frac{1}{6350} a^{8} + \frac{11}{50} a^{5} - \frac{21}{50} a^{2}$, $\frac{1}{117576377750} a^{9} - \frac{655132}{462899125} a^{6} - \frac{178598448}{462899125} a^{3} + \frac{153}{58318}$, $\frac{1}{587881888750} a^{10} - \frac{655132}{2314495625} a^{7} - \frac{1104396698}{2314495625} a^{4} - \frac{11633}{58318} a$, $\frac{1}{373304999356250} a^{11} + \frac{2334611}{2939409443750} a^{8} - \frac{24890850521}{2939409443750} a^{5} - \frac{32118562}{92579825} a^{2}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{48}\times C_{1680}$, which has order $1290240$ (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:  \( \frac{2}{35607625} a^{9} - \frac{1931}{35607625} a^{6} + \frac{1708}{280375} a^{3} - \frac{3755}{2243} \) (order $6$)
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:  \( 60076.605409926466 \) (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

\(\Q(\sqrt{-127}) \), \(\Q(\sqrt{381}) \), \(\Q(\sqrt{-3}) \), 3.3.1306449.2, \(\Q(\sqrt{-3}, \sqrt{-127})\), 6.0.216764741679327.2, 6.6.650294225037981.2, 6.0.5120426968803.4

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.18.51$x^{12} + 27 x^{11} + 15 x^{10} + 36 x^{9} - 36 x^{8} - 18 x^{7} + 21 x^{6} - 18 x^{4} + 27 x^{2} - 27 x + 36$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$127$127.6.5.5$x^{6} + 92583$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.5$x^{6} + 92583$$6$$1$$5$$C_6$$[\ ]_{6}$