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Properties

Label 12.0.17039290996...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 73^{6}$
Root discriminant $104.54$
Ramified primes $5, 7, 73$
Class number $120250$ (GRH)
Class group $[5, 5, 4810]$ (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![6145345751, 233929177, 866225088, 23582808, 51291336, 768444, 1619354, 5828, 28463, -81, 263, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 263*x^10 - 81*x^9 + 28463*x^8 + 5828*x^7 + 1619354*x^6 + 768444*x^5 + 51291336*x^4 + 23582808*x^3 + 866225088*x^2 + 233929177*x + 6145345751)
 
gp: K = bnfinit(x^12 - x^11 + 263*x^10 - 81*x^9 + 28463*x^8 + 5828*x^7 + 1619354*x^6 + 768444*x^5 + 51291336*x^4 + 23582808*x^3 + 866225088*x^2 + 233929177*x + 6145345751, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 263 x^{10} - 81 x^{9} + 28463 x^{8} + 5828 x^{7} + 1619354 x^{6} + 768444 x^{5} + 51291336 x^{4} + 23582808 x^{3} + 866225088 x^{2} + 233929177 x + 6145345751 \)

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:  \(1703929099697370095703125=5^{9}\cdot 7^{8}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.54$
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:  $5, 7, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2555=5\cdot 7\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{2555}(1,·)$, $\chi_{2555}(802,·)$, $\chi_{2555}(583,·)$, $\chi_{2555}(72,·)$, $\chi_{2555}(74,·)$, $\chi_{2555}(1899,·)$, $\chi_{2555}(1313,·)$, $\chi_{2555}(366,·)$, $\chi_{2555}(1096,·)$, $\chi_{2555}(2262,·)$, $\chi_{2555}(218,·)$, $\chi_{2555}(1534,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{379} a^{9} + \frac{54}{379} a^{8} - \frac{47}{379} a^{7} - \frac{85}{379} a^{6} + \frac{135}{379} a^{5} - \frac{88}{379} a^{4} + \frac{98}{379} a^{3} - \frac{18}{379} a^{2} - \frac{64}{379} a + \frac{182}{379}$, $\frac{1}{1103269} a^{10} + \frac{289}{1103269} a^{9} + \frac{202143}{1103269} a^{8} + \frac{520228}{1103269} a^{7} - \frac{154385}{1103269} a^{6} + \frac{2047}{15539} a^{5} - \frac{128218}{1103269} a^{4} - \frac{80455}{1103269} a^{3} - \frac{356006}{1103269} a^{2} - \frac{311994}{1103269} a - \frac{432875}{1103269}$, $\frac{1}{929540747578317015287435517821699} a^{11} + \frac{257249171993851439069277879}{929540747578317015287435517821699} a^{10} - \frac{679137603133279111906940862933}{929540747578317015287435517821699} a^{9} + \frac{87800826313240006648107972619742}{929540747578317015287435517821699} a^{8} - \frac{216019552293354182507874257801695}{929540747578317015287435517821699} a^{7} + \frac{14521461015661557095224701444298}{929540747578317015287435517821699} a^{6} + \frac{110922639008885370274392865041024}{929540747578317015287435517821699} a^{5} - \frac{233251802489895755909442404589007}{929540747578317015287435517821699} a^{4} + \frac{181694103315895234008940249579221}{929540747578317015287435517821699} a^{3} - \frac{300838771709273441825150808570469}{929540747578317015287435517821699} a^{2} + \frac{373815484221374780892676119008165}{929540747578317015287435517821699} a + \frac{52471987561253113629500920975566}{929540747578317015287435517821699}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{4810}$, which has order $120250$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.666125.1, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$73$73.12.6.2$x^{12} - 2073071593 x^{2} + 756671131445$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$