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Properties

Label 12.0.45006250477...1152.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 13^{11}\cdot 19^{10}$
Root discriminant $244.22$
Ramified primes $2, 13, 19$
Class number $1210800$ (GRH)
Class group $[2, 2, 10, 30270]$ (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![816061077, 0, 296048519, 0, 27702779, 0, 1147562, 0, 23959, 0, 247, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 247*x^10 + 23959*x^8 + 1147562*x^6 + 27702779*x^4 + 296048519*x^2 + 816061077)
 
gp: K = bnfinit(x^12 + 247*x^10 + 23959*x^8 + 1147562*x^6 + 27702779*x^4 + 296048519*x^2 + 816061077, 1)
 

Normalized defining polynomial

\( x^{12} + 247 x^{10} + 23959 x^{8} + 1147562 x^{6} + 27702779 x^{4} + 296048519 x^{2} + 816061077 \)

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:  \(45006250477353349878303281152=2^{12}\cdot 13^{11}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $244.22$
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:  $2, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(988=2^{2}\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{988}(1,·)$, $\chi_{988}(483,·)$, $\chi_{988}(487,·)$, $\chi_{988}(809,·)$, $\chi_{988}(759,·)$, $\chi_{988}(77,·)$, $\chi_{988}(943,·)$, $\chi_{988}(49,·)$, $\chi_{988}(425,·)$, $\chi_{988}(121,·)$, $\chi_{988}(635,·)$, $\chi_{988}(151,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{2052} a^{6} - \frac{2}{27} a^{4} - \frac{5}{27} a^{2} - \frac{1}{12}$, $\frac{1}{2052} a^{7} - \frac{2}{27} a^{5} + \frac{4}{27} a^{3} - \frac{5}{12} a$, $\frac{1}{147744} a^{8} - \frac{5}{49248} a^{6} - \frac{19}{216} a^{4} + \frac{3407}{7776} a^{2} + \frac{67}{864}$, $\frac{1}{147744} a^{9} - \frac{5}{49248} a^{7} - \frac{19}{216} a^{5} + \frac{815}{7776} a^{3} + \frac{355}{864} a$, $\frac{1}{5318784} a^{10} + \frac{1}{2659392} a^{8} - \frac{97}{1772928} a^{6} + \frac{43619}{279936} a^{4} + \frac{59069}{139968} a^{2} + \frac{13235}{31104}$, $\frac{1}{739310976} a^{11} + \frac{1027}{369655488} a^{9} + \frac{24203}{246436992} a^{7} + \frac{983219}{38911104} a^{5} + \frac{1678043}{19455552} a^{3} - \frac{2016193}{4323456} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{30270}$, which has order $1210800$ (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:  \( 288307.7204812408 \) (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

\(\Q(\sqrt{13}) \), 3.3.61009.1, 4.0.12689872.1, 6.6.48387275053.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$13$13.12.11.3$x^{12} - 208$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$19$19.12.10.6$x^{12} - 209 x^{6} + 11552$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$