↑

Properties

Label 12.0.31643295525...0625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 5^{6}\cdot 19^{6}$
Root discriminant $42.17$
Ramified primes $3, 5, 19$
Class number $192$
Class group $[2, 2, 2, 2, 12]$
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![156329, 5118, 77817, 5580, 17949, 1200, 2281, 36, 183, -4, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 9*x^10 - 4*x^9 + 183*x^8 + 36*x^7 + 2281*x^6 + 1200*x^5 + 17949*x^4 + 5580*x^3 + 77817*x^2 + 5118*x + 156329)
 
gp: K = bnfinit(x^12 + 9*x^10 - 4*x^9 + 183*x^8 + 36*x^7 + 2281*x^6 + 1200*x^5 + 17949*x^4 + 5580*x^3 + 77817*x^2 + 5118*x + 156329, 1)
 

Normalized defining polynomial

\( x^{12} + 9 x^{10} - 4 x^{9} + 183 x^{8} + 36 x^{7} + 2281 x^{6} + 1200 x^{5} + 17949 x^{4} + 5580 x^{3} + 77817 x^{2} + 5118 x + 156329 \)

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:  \(31643295525096890625=3^{16}\cdot 5^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.17$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(855=3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{855}(1,·)$, $\chi_{855}(514,·)$, $\chi_{855}(379,·)$, $\chi_{855}(229,·)$, $\chi_{855}(721,·)$, $\chi_{855}(436,·)$, $\chi_{855}(286,·)$, $\chi_{855}(151,·)$, $\chi_{855}(664,·)$, $\chi_{855}(571,·)$, $\chi_{855}(94,·)$, $\chi_{855}(799,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{36} a^{9} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{5}{12} a^{4} - \frac{17}{36} a^{3} + \frac{1}{6} a^{2} - \frac{1}{4} a - \frac{17}{36}$, $\frac{1}{539172} a^{10} - \frac{851}{134793} a^{9} - \frac{7487}{44931} a^{8} + \frac{907}{59908} a^{7} - \frac{12133}{59908} a^{6} - \frac{66049}{179724} a^{5} - \frac{133493}{539172} a^{4} + \frac{1901}{269586} a^{3} - \frac{78427}{179724} a^{2} - \frac{37979}{539172} a + \frac{82961}{269586}$, $\frac{1}{9454492628604} a^{11} + \frac{1874009}{4727246314302} a^{10} - \frac{39546035107}{4727246314302} a^{9} + \frac{207276782795}{1050499180956} a^{8} + \frac{488867021281}{3151497542868} a^{7} + \frac{7569332885}{185382208404} a^{6} - \frac{3951773005763}{9454492628604} a^{5} + \frac{246000987154}{2363623157151} a^{4} - \frac{98404806737}{9454492628604} a^{3} - \frac{4451751483167}{9454492628604} a^{2} - \frac{623921877503}{2363623157151} a + \frac{1797585913805}{4727246314302}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $192$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 201.000834787 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-19})\), 6.6.820125.1, 6.0.5625237375.5, 6.0.45001899.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$