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Properties

Label 12.0.81672801965...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{10}\cdot 23^{6}$
Root discriminant $81.16$
Ramified primes $5, 7, 23$
Class number $12500$ (GRH)
Class group $[5, 50, 50]$ (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![324352981, -73475842, 82172540, -10507816, 9098780, -554959, 508507, -14743, 14544, -199, 198, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 198*x^10 - 199*x^9 + 14544*x^8 - 14743*x^7 + 508507*x^6 - 554959*x^5 + 9098780*x^4 - 10507816*x^3 + 82172540*x^2 - 73475842*x + 324352981)
 
gp: K = bnfinit(x^12 - x^11 + 198*x^10 - 199*x^9 + 14544*x^8 - 14743*x^7 + 508507*x^6 - 554959*x^5 + 9098780*x^4 - 10507816*x^3 + 82172540*x^2 - 73475842*x + 324352981, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 198 x^{10} - 199 x^{9} + 14544 x^{8} - 14743 x^{7} + 508507 x^{6} - 554959 x^{5} + 9098780 x^{4} - 10507816 x^{3} + 82172540 x^{2} - 73475842 x + 324352981 \)

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:  \(81672801965256564453125=5^{9}\cdot 7^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.16$
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:  $5, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(805=5\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{805}(576,·)$, $\chi_{805}(1,·)$, $\chi_{805}(482,·)$, $\chi_{805}(643,·)$, $\chi_{805}(68,·)$, $\chi_{805}(712,·)$, $\chi_{805}(367,·)$, $\chi_{805}(528,·)$, $\chi_{805}(116,·)$, $\chi_{805}(599,·)$, $\chi_{805}(484,·)$, $\chi_{805}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{14726039} a^{10} + \frac{3615354}{14726039} a^{9} - \frac{5428791}{14726039} a^{8} - \frac{6616262}{14726039} a^{7} - \frac{3323304}{14726039} a^{6} + \frac{2869890}{14726039} a^{5} - \frac{5016410}{14726039} a^{4} - \frac{1800638}{14726039} a^{3} - \frac{3654589}{14726039} a^{2} - \frac{5119214}{14726039} a - \frac{578871}{14726039}$, $\frac{1}{16916673699504424342261128638779} a^{11} + \frac{371390851267667211848760}{16916673699504424342261128638779} a^{10} + \frac{2962314302647762132409023135106}{16916673699504424342261128638779} a^{9} + \frac{3042992314208291145403597281326}{16916673699504424342261128638779} a^{8} - \frac{454284902283822860377511328067}{16916673699504424342261128638779} a^{7} - \frac{7341962953781893507633224078254}{16916673699504424342261128638779} a^{6} + \frac{6082765487427102128448152708475}{16916673699504424342261128638779} a^{5} + \frac{7875559493493821589021827230705}{16916673699504424342261128638779} a^{4} + \frac{380654766771596338834088808525}{16916673699504424342261128638779} a^{3} + \frac{4295492323623615918001256319515}{16916673699504424342261128638779} a^{2} - \frac{6711669495841765283393365335099}{16916673699504424342261128638779} a - \frac{18549553015022176937920}{52155135578989558730959}$

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

Class group and class number

$C_{5}\times C_{50}\times C_{50}$, which has order $12500$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.3240125.1, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$23$23.12.6.2$x^{12} - 6436343 x^{2} + 2220538335$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$