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Properties

Label 12.0.10331448031...1637.2
Degree $12$
Signature $[0, 6]$
Discriminant $7^{8}\cdot 13^{11}$
Root discriminant $38.42$
Ramified primes $7, 13$
Class number $111$
Class group $[111]$
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4941, -2133, 1782, -573, 131, 649, 326, -183, 27, -27, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + x^10 - 27*x^9 + 27*x^8 - 183*x^7 + 326*x^6 + 649*x^5 + 131*x^4 - 573*x^3 + 1782*x^2 - 2133*x + 4941)
 
gp: K = bnfinit(x^12 - x^11 + x^10 - 27*x^9 + 27*x^8 - 183*x^7 + 326*x^6 + 649*x^5 + 131*x^4 - 573*x^3 + 1782*x^2 - 2133*x + 4941, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} - 183 x^{7} + 326 x^{6} + 649 x^{5} + 131 x^{4} - 573 x^{3} + 1782 x^{2} - 2133 x + 4941 \)

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:  \(10331448031704891637=7^{8}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.42$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(67,·)$, $\chi_{91}(8,·)$, $\chi_{91}(9,·)$, $\chi_{91}(11,·)$, $\chi_{91}(81,·)$, $\chi_{91}(88,·)$, $\chi_{91}(72,·)$, $\chi_{91}(57,·)$, $\chi_{91}(58,·)$, $\chi_{91}(30,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{4} - \frac{7}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{243} a^{10} + \frac{1}{243} a^{9} - \frac{1}{81} a^{7} - \frac{2}{81} a^{6} + \frac{10}{81} a^{5} - \frac{28}{243} a^{4} - \frac{10}{243} a^{3} - \frac{25}{81} a^{2} + \frac{7}{27} a - \frac{2}{9}$, $\frac{1}{584514010107} a^{11} + \frac{102820490}{584514010107} a^{10} - \frac{3525995738}{584514010107} a^{9} - \frac{911944600}{194838003369} a^{8} - \frac{5822168}{801802483} a^{7} - \frac{3802366750}{194838003369} a^{6} + \frac{56952558452}{584514010107} a^{5} + \frac{96646073356}{584514010107} a^{4} + \frac{41936861690}{584514010107} a^{3} + \frac{18673231442}{194838003369} a^{2} + \frac{13554309418}{64946001123} a + \frac{51729076}{354896181}$

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

Class group and class number

$C_{111}$, which has order $111$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17569.0378451 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.8281.2, 4.0.2197.1, 6.6.891474493.2

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$7.12.8.3$x^{12} + 14 x^{9} + 539 x^{6} + 343 x^{3} + 60025$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$