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Properties

Label 12.0.87726396011...5712.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 7^{8}\cdot 11^{6}$
Root discriminant $81.64$
Ramified primes $2, 7, 11$
Class number $14130$ (GRH)
Class group $[3, 4710]$ (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![31017889, -7513800, 10909864, -2100652, 1713002, -285164, 155274, -16420, 6567, -416, 130, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 130*x^10 - 416*x^9 + 6567*x^8 - 16420*x^7 + 155274*x^6 - 285164*x^5 + 1713002*x^4 - 2100652*x^3 + 10909864*x^2 - 7513800*x + 31017889)
 
gp: K = bnfinit(x^12 - 4*x^11 + 130*x^10 - 416*x^9 + 6567*x^8 - 16420*x^7 + 155274*x^6 - 285164*x^5 + 1713002*x^4 - 2100652*x^3 + 10909864*x^2 - 7513800*x + 31017889, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 130 x^{10} - 416 x^{9} + 6567 x^{8} - 16420 x^{7} + 155274 x^{6} - 285164 x^{5} + 1713002 x^{4} - 2100652 x^{3} + 10909864 x^{2} - 7513800 x + 31017889 \)

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:  \(87726396011200183795712=2^{33}\cdot 7^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.64$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1232=2^{4}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1232}(1,·)$, $\chi_{1232}(197,·)$, $\chi_{1232}(529,·)$, $\chi_{1232}(617,·)$, $\chi_{1232}(109,·)$, $\chi_{1232}(813,·)$, $\chi_{1232}(177,·)$, $\chi_{1232}(725,·)$, $\chi_{1232}(793,·)$, $\chi_{1232}(1145,·)$, $\chi_{1232}(989,·)$, $\chi_{1232}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} - \frac{5}{11} a^{3} + \frac{2}{11} a^{2} - \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{7} + \frac{4}{11} a^{5} + \frac{3}{11} a^{3} + \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{8} - \frac{3}{11} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{9} - \frac{2}{11} a^{5} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{28670653099} a^{10} + \frac{638354004}{28670653099} a^{9} - \frac{669756537}{28670653099} a^{8} + \frac{796553822}{28670653099} a^{7} - \frac{1268776165}{28670653099} a^{6} - \frac{6575003636}{28670653099} a^{5} + \frac{13825521669}{28670653099} a^{4} + \frac{12780364825}{28670653099} a^{3} - \frac{1823769778}{28670653099} a^{2} - \frac{7171093034}{28670653099} a + \frac{7442620925}{28670653099}$, $\frac{1}{2953023437666142420283} a^{11} - \frac{21019776408}{2953023437666142420283} a^{10} - \frac{57234305295853836056}{2953023437666142420283} a^{9} + \frac{40346598622658689996}{2953023437666142420283} a^{8} - \frac{88922581774246982544}{2953023437666142420283} a^{7} - \frac{40784669442460789122}{2953023437666142420283} a^{6} + \frac{1055642833476005573973}{2953023437666142420283} a^{5} - \frac{406172383954321136045}{2953023437666142420283} a^{4} - \frac{536365925249787643400}{2953023437666142420283} a^{3} + \frac{1464430792608602872333}{2953023437666142420283} a^{2} + \frac{627743772819920506826}{2953023437666142420283} a - \frac{211326712827821318015}{2953023437666142420283}$

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

Class group and class number

$C_{3}\times C_{4710}$, which has order $14130$ (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:  \( 279.150027194 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.0.247808.2, 6.6.1229312.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$2$2.12.33.374$x^{12} + 28 x^{10} - 6 x^{8} + 40 x^{6} - 56 x^{4} - 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$11$11.12.6.2$x^{12} + 14641 x^{4} - 322102 x^{2} + 14172488$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$