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Properties

Label 12.0.45641840196...2209.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 53^{6}$
Root discriminant $63.82$
Ramified primes $3, 7, 53$
Class number $5880$ (GRH)
Class group $[7, 840]$ (GRH)
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![12660103, 6541378, 5432804, 755704, 1664623, -19929, 168806, -3276, 7396, -103, 143, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 143*x^10 - 103*x^9 + 7396*x^8 - 3276*x^7 + 168806*x^6 - 19929*x^5 + 1664623*x^4 + 755704*x^3 + 5432804*x^2 + 6541378*x + 12660103)
 
gp: K = bnfinit(x^12 - x^11 + 143*x^10 - 103*x^9 + 7396*x^8 - 3276*x^7 + 168806*x^6 - 19929*x^5 + 1664623*x^4 + 755704*x^3 + 5432804*x^2 + 6541378*x + 12660103, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 143 x^{10} - 103 x^{9} + 7396 x^{8} - 3276 x^{7} + 168806 x^{6} - 19929 x^{5} + 1664623 x^{4} + 755704 x^{3} + 5432804 x^{2} + 6541378 x + 12660103 \)

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:  \(4564184019624502972209=3^{6}\cdot 7^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.82$
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:  $3, 7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1113=3\cdot 7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{1113}(1,·)$, $\chi_{1113}(902,·)$, $\chi_{1113}(584,·)$, $\chi_{1113}(425,·)$, $\chi_{1113}(1006,·)$, $\chi_{1113}(370,·)$, $\chi_{1113}(52,·)$, $\chi_{1113}(158,·)$, $\chi_{1113}(953,·)$, $\chi_{1113}(317,·)$, $\chi_{1113}(478,·)$, $\chi_{1113}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3277} a^{7} + \frac{91}{3277} a^{5} - \frac{911}{3277} a^{3} - \frac{1006}{3277} a + \frac{726}{3277}$, $\frac{1}{3277} a^{8} + \frac{91}{3277} a^{6} - \frac{911}{3277} a^{4} - \frac{1006}{3277} a^{2} + \frac{726}{3277} a$, $\frac{1}{45878} a^{9} + \frac{5}{45878} a^{7} - \frac{1}{14} a^{6} - \frac{8737}{45878} a^{5} + \frac{3}{7} a^{4} + \frac{15077}{45878} a^{3} - \frac{22213}{45878} a^{2} - \frac{9174}{22939} a + \frac{19489}{45878}$, $\frac{1}{1248232520822} a^{10} + \frac{6005295}{1248232520822} a^{9} - \frac{73138823}{1248232520822} a^{8} + \frac{66360106}{624116260411} a^{7} - \frac{242176915951}{624116260411} a^{6} + \frac{114342501557}{1248232520822} a^{5} + \frac{291209631311}{1248232520822} a^{4} - \frac{267363368841}{624116260411} a^{3} + \frac{418036904357}{1248232520822} a^{2} - \frac{331284868993}{1248232520822} a + \frac{299418114921}{1248232520822}$, $\frac{1}{1248232520822} a^{11} - \frac{5628472}{624116260411} a^{9} + \frac{188992257}{1248232520822} a^{8} - \frac{206690}{89159465773} a^{7} + \frac{431909527087}{1248232520822} a^{6} - \frac{109138145587}{624116260411} a^{5} + \frac{474667468851}{1248232520822} a^{4} + \frac{288382173439}{1248232520822} a^{3} + \frac{199600911070}{624116260411} a^{2} + \frac{194900952486}{624116260411} a - \frac{53412018909}{1248232520822}$

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

Class group and class number

$C_{7}\times C_{840}$, which has order $5880$ (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:  \( 140.7987960054707 \) (assuming GRH)
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

\(\Q(\sqrt{-371}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-159}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-159})\), 6.0.2502175739.2, \(\Q(\zeta_{21})^+\), 6.0.9651249279.4

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$53$53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$