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Properties

Label 12.0.45981932521...3133.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{11}\cdot 37^{6}$
Root discriminant $63.86$
Ramified primes $13, 37$
Class number $3650$ (GRH)
Class group $[3650]$ (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![13595401, -6686668, 6686668, -1313209, 1313209, -119107, 119107, -5383, 5383, -118, 118, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 118*x^10 - 118*x^9 + 5383*x^8 - 5383*x^7 + 119107*x^6 - 119107*x^5 + 1313209*x^4 - 1313209*x^3 + 6686668*x^2 - 6686668*x + 13595401)
 
gp: K = bnfinit(x^12 - x^11 + 118*x^10 - 118*x^9 + 5383*x^8 - 5383*x^7 + 119107*x^6 - 119107*x^5 + 1313209*x^4 - 1313209*x^3 + 6686668*x^2 - 6686668*x + 13595401, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 118 x^{10} - 118 x^{9} + 5383 x^{8} - 5383 x^{7} + 119107 x^{6} - 119107 x^{5} + 1313209 x^{4} - 1313209 x^{3} + 6686668 x^{2} - 6686668 x + 13595401 \)

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:  \(4598193252144577023133=13^{11}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.86$
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:  $13, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(481=13\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{481}(1,·)$, $\chi_{481}(258,·)$, $\chi_{481}(38,·)$, $\chi_{481}(73,·)$, $\chi_{481}(75,·)$, $\chi_{481}(332,·)$, $\chi_{481}(334,·)$, $\chi_{481}(369,·)$, $\chi_{481}(110,·)$, $\chi_{481}(184,·)$, $\chi_{481}(186,·)$, $\chi_{481}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2295721} a^{7} - \frac{1139666}{2295721} a^{6} + \frac{63}{2295721} a^{5} + \frac{442503}{2295721} a^{4} + \frac{1134}{2295721} a^{3} + \frac{234488}{2295721} a^{2} + \frac{5103}{2295721} a + \frac{468976}{2295721}$, $\frac{1}{2295721} a^{8} + \frac{72}{2295721} a^{6} + \frac{1074110}{2295721} a^{5} + \frac{1620}{2295721} a^{4} + \frac{124809}{2295721} a^{3} + \frac{11664}{2295721} a^{2} + \frac{1123281}{2295721} a + \frac{13122}{2295721}$, $\frac{1}{2295721} a^{9} + \frac{484106}{2295721} a^{6} - \frac{2916}{2295721} a^{5} + \frac{404687}{2295721} a^{4} - \frac{69984}{2295721} a^{3} + \frac{310192}{2295721} a^{2} - \frac{354294}{2295721} a + \frac{669543}{2295721}$, $\frac{1}{2295721} a^{10} - \frac{3645}{2295721} a^{6} - \frac{249618}{2295721} a^{5} - \frac{109350}{2295721} a^{4} + \frac{11307}{2295721} a^{3} - \frac{885735}{2295721} a^{2} + \frac{472421}{2295721} a - \frac{1062882}{2295721}$, $\frac{1}{2295721} a^{11} + \frac{922822}{2295721} a^{6} + \frac{120285}{2295721} a^{5} - \frac{957121}{2295721} a^{4} + \frac{951974}{2295721} a^{3} - \frac{1122752}{2295721} a^{2} - \frac{828215}{2295721} a - \frac{894625}{2295721}$

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

Class group and class number

$C_{3650}$, which has order $3650$ (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:  \( -1 \) (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:  \( 120.784031363 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.3007693.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$37$37.12.6.2$x^{12} - 69343957 x^{2} + 51314528180$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$