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Properties

Label 12.0.97080386290...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{9}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $120.85$
Ramified primes $3, 5, 7, 17$
Class number $143624$ (GRH)
Class group $[2, 2, 35906]$ (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![31633408171, -4044668347, 4246142315, -256778386, 223464715, -6113764, 5685112, -73298, 72854, -444, 443, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171)
 
gp: K = bnfinit(x^12 - x^11 + 443*x^10 - 444*x^9 + 72854*x^8 - 73298*x^7 + 5685112*x^6 - 6113764*x^5 + 223464715*x^4 - 256778386*x^3 + 4246142315*x^2 - 4044668347*x + 31633408171, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 443 x^{10} - 444 x^{9} + 72854 x^{8} - 73298 x^{7} + 5685112 x^{6} - 6113764 x^{5} + 223464715 x^{4} - 256778386 x^{3} + 4246142315 x^{2} - 4044668347 x + 31633408171 \)

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:  \(9708038629029565330078125=3^{6}\cdot 5^{9}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.85$
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:  $3, 5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1785=3\cdot 5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1785}(256,·)$, $\chi_{1785}(1,·)$, $\chi_{1785}(1223,·)$, $\chi_{1785}(713,·)$, $\chi_{1785}(458,·)$, $\chi_{1785}(1427,·)$, $\chi_{1785}(1172,·)$, $\chi_{1785}(1429,·)$, $\chi_{1785}(919,·)$, $\chi_{1785}(152,·)$, $\chi_{1785}(1684,·)$, $\chi_{1785}(1276,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1024846549} a^{10} + \frac{114206734}{1024846549} a^{9} - \frac{171396636}{1024846549} a^{8} + \frac{108291908}{1024846549} a^{7} + \frac{41425790}{1024846549} a^{6} + \frac{102298999}{1024846549} a^{5} - \frac{475518299}{1024846549} a^{4} - \frac{117684046}{1024846549} a^{3} + \frac{96287745}{1024846549} a^{2} - \frac{42480026}{1024846549} a + \frac{387181469}{1024846549}$, $\frac{1}{589945208928960574184455194739022010221} a^{11} + \frac{263135947823757304653226592045}{589945208928960574184455194739022010221} a^{10} + \frac{194638854115484710702628655110030062178}{589945208928960574184455194739022010221} a^{9} + \frac{107028742041971317496752959457457020185}{589945208928960574184455194739022010221} a^{8} + \frac{246282536755264633244784906464111582432}{589945208928960574184455194739022010221} a^{7} - \frac{1151411631800253320097798602336171620}{8309087449703670058935988658296084651} a^{6} + \frac{2554640049685038694130548367434644855}{8309087449703670058935988658296084651} a^{5} - \frac{270942970524802128092857516202575364539}{589945208928960574184455194739022010221} a^{4} - \frac{25917515365986797778296140109137593777}{589945208928960574184455194739022010221} a^{3} + \frac{171008037141005076166014150380005653963}{589945208928960574184455194739022010221} a^{2} + \frac{253413893355046561928190069401236806748}{589945208928960574184455194739022010221} a - \frac{3835831337866137457383294008}{18649435613763369543708948551}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{35906}$, which has order $143624$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.15931125.1, 6.6.300125.1

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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$17$17.12.6.2$x^{12} - 1419857 x^{2} + 289650828$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$