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Properties

Label 12.0.500422134593689.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{10}\cdot 11^{6}$
Root discriminant $16.79$
Ramified primes $7, 11$
Class number $1$
Class group Trivial
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![729, -243, -162, 135, 9, -48, 13, -16, 1, 5, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 2*x^10 + 5*x^9 + x^8 - 16*x^7 + 13*x^6 - 48*x^5 + 9*x^4 + 135*x^3 - 162*x^2 - 243*x + 729)
 
gp: K = bnfinit(x^12 - x^11 - 2*x^10 + 5*x^9 + x^8 - 16*x^7 + 13*x^6 - 48*x^5 + 9*x^4 + 135*x^3 - 162*x^2 - 243*x + 729, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 2 x^{10} + 5 x^{9} + x^{8} - 16 x^{7} + 13 x^{6} - 48 x^{5} + 9 x^{4} + 135 x^{3} - 162 x^{2} - 243 x + 729 \)

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:  \(500422134593689=7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.79$
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:  $7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(32,·)$, $\chi_{77}(1,·)$, $\chi_{77}(34,·)$, $\chi_{77}(67,·)$, $\chi_{77}(65,·)$, $\chi_{77}(12,·)$, $\chi_{77}(10,·)$, $\chi_{77}(43,·)$, $\chi_{77}(76,·)$, $\chi_{77}(45,·)$, $\chi_{77}(54,·)$, $\chi_{77}(23,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{39} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{3}{13}$, $\frac{1}{117} a^{8} - \frac{1}{117} a^{7} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{16}{39} a + \frac{1}{13}$, $\frac{1}{351} a^{9} - \frac{1}{351} a^{8} - \frac{2}{351} a^{7} - \frac{10}{27} a^{6} - \frac{2}{27} a^{5} + \frac{5}{27} a^{4} + \frac{1}{27} a^{3} - \frac{16}{117} a^{2} + \frac{1}{39} a + \frac{5}{13}$, $\frac{1}{1053} a^{10} - \frac{1}{1053} a^{9} - \frac{2}{1053} a^{8} + \frac{5}{1053} a^{7} + \frac{25}{81} a^{6} + \frac{5}{81} a^{5} + \frac{1}{81} a^{4} - \frac{16}{351} a^{3} + \frac{1}{117} a^{2} + \frac{5}{39} a - \frac{2}{13}$, $\frac{1}{3159} a^{11} - \frac{1}{3159} a^{10} - \frac{2}{3159} a^{9} + \frac{5}{3159} a^{8} + \frac{1}{3159} a^{7} - \frac{76}{243} a^{6} + \frac{1}{243} a^{5} - \frac{16}{1053} a^{4} + \frac{1}{351} a^{3} + \frac{5}{117} a^{2} - \frac{2}{39} a - \frac{1}{13}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1}{3159} a^{11} + \frac{2}{3159} a^{10} - \frac{5}{3159} a^{9} - \frac{1}{3159} a^{8} + \frac{16}{3159} a^{7} - \frac{1}{243} a^{6} + \frac{16}{243} a^{5} - \frac{1}{351} a^{4} - \frac{5}{117} a^{3} + \frac{2}{39} a^{2} + \frac{1}{13} a - \frac{3}{13} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 706.974937058 \)
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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-7}, \sqrt{-11})\), 6.6.22370117.1, 6.0.3195731.1, \(\Q(\zeta_{7})\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$