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Properties

Label 12.12.3443737680...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{12}\cdot 3^{16}\cdot 5^{9}$
Root discriminant $28.93$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, 42, -96, -336, 795, 72, -737, 36, 234, -4, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 27*x^10 - 4*x^9 + 234*x^8 + 36*x^7 - 737*x^6 + 72*x^5 + 795*x^4 - 336*x^3 - 96*x^2 + 42*x + 1)
 
gp: K = bnfinit(x^12 - 27*x^10 - 4*x^9 + 234*x^8 + 36*x^7 - 737*x^6 + 72*x^5 + 795*x^4 - 336*x^3 - 96*x^2 + 42*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1 \)

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:  \(344373768000000000=2^{12}\cdot 3^{16}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.93$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(180=2^{2}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(163,·)$, $\chi_{180}(103,·)$, $\chi_{180}(169,·)$, $\chi_{180}(7,·)$, $\chi_{180}(109,·)$, $\chi_{180}(49,·)$, $\chi_{180}(67,·)$, $\chi_{180}(43,·)$, $\chi_{180}(121,·)$, $\chi_{180}(61,·)$, $\chi_{180}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3401} a^{10} + \frac{1219}{3401} a^{9} + \frac{1259}{3401} a^{8} + \frac{384}{3401} a^{7} - \frac{12}{179} a^{6} - \frac{1183}{3401} a^{5} + \frac{1}{19} a^{4} - \frac{26}{179} a^{3} + \frac{1302}{3401} a^{2} - \frac{1066}{3401} a - \frac{442}{3401}$, $\frac{1}{17205659} a^{11} + \frac{1694}{17205659} a^{10} - \frac{8007241}{17205659} a^{9} + \frac{475973}{17205659} a^{8} - \frac{32477}{905561} a^{7} + \frac{8505250}{17205659} a^{6} - \frac{8142575}{17205659} a^{5} + \frac{152482}{905561} a^{4} + \frac{1456949}{17205659} a^{3} - \frac{4545334}{17205659} a^{2} + \frac{6937997}{17205659} a - \frac{20537}{905561}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 66792.5779024 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{20})^+\), 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 R R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_5.4t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{4} - 5 x^{2} + 5$ $C_4$ (as 4T1) $0$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2_5.4t1.1c2$1$ $ 2^{2} \cdot 5 $ $x^{4} - 5 x^{2} + 5$ $C_4$ (as 4T1) $0$ $1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2_5.12t1.1c1$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1$ $C_{12}$ (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) $0$ $1$
* 1.2e2_3e2_5.12t1.1c2$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1$ $C_{12}$ (as 12T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2_5.12t1.1c3$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1$ $C_{12}$ (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) $0$ $1$
* 1.2e2_3e2_5.12t1.1c4$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1$ $C_{12}$ (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 *.