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Properties

Label 12.12.756680642578125.1
Degree $12$
Signature $[12, 0]$
Discriminant $3^{18}\cdot 5^{9}$
Root discriminant $17.37$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
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![1, -12, -36, 31, 105, -27, -112, 9, 54, -1, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - x^9 + 54*x^8 + 9*x^7 - 112*x^6 - 27*x^5 + 105*x^4 + 31*x^3 - 36*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^12 - 12*x^10 - x^9 + 54*x^8 + 9*x^7 - 112*x^6 - 27*x^5 + 105*x^4 + 31*x^3 - 36*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(756680642578125=3^{18}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.37$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(45=3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{45}(32,·)$, $\chi_{45}(1,·)$, $\chi_{45}(2,·)$, $\chi_{45}(4,·)$, $\chi_{45}(38,·)$, $\chi_{45}(8,·)$, $\chi_{45}(34,·)$, $\chi_{45}(16,·)$, $\chi_{45}(17,·)$, $\chi_{45}(19,·)$, $\chi_{45}(23,·)$, $\chi_{45}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$

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:  $11$
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:  \( 2075.03148059 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{15})^+\), 6.6.820125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_5.4t1.1c1$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.3_5.4t1.1c2$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.12t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1$ $C_{12}$ (as 12T1) $0$ $1$
* 1.3e2_5.6t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2_5.12t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1$ $C_{12}$ (as 12T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.12t1.1c3$1$ $ 3^{2} \cdot 5 $ $x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1$ $C_{12}$ (as 12T1) $0$ $1$
* 1.3e2_5.6t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2_5.12t1.1c4$1$ $ 3^{2} \cdot 5 $ $x^{12} - 12 x^{10} - x^{9} + 54 x^{8} + 9 x^{7} - 112 x^{6} - 27 x^{5} + 105 x^{4} + 31 x^{3} - 36 x^{2} - 12 x + 1$ $C_{12}$ (as 12T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.