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Properties

Label 12.0.19795224397...5648.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{18}\cdot 29^{6}$
Root discriminant $188.24$
Ramified primes $2, 3, 29$
Class number $1066468$ (GRH)
Class group $[2, 533234]$ (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![42827279112, 0, 8860816368, 0, 534704436, 0, 13462728, 0, 151380, 0, 696, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112)
 
gp: K = bnfinit(x^12 + 696*x^10 + 151380*x^8 + 13462728*x^6 + 534704436*x^4 + 8860816368*x^2 + 42827279112, 1)
 

Normalized defining polynomial

\( x^{12} + 696 x^{10} + 151380 x^{8} + 13462728 x^{6} + 534704436 x^{4} + 8860816368 x^{2} + 42827279112 \)

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:  \(1979522439778248324859035648=2^{33}\cdot 3^{18}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.24$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4176=2^{4}\cdot 3^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4176}(1,·)$, $\chi_{4176}(3653,·)$, $\chi_{4176}(2785,·)$, $\chi_{4176}(2089,·)$, $\chi_{4176}(2957,·)$, $\chi_{4176}(173,·)$, $\chi_{4176}(1393,·)$, $\chi_{4176}(2261,·)$, $\chi_{4176}(3481,·)$, $\chi_{4176}(697,·)$, $\chi_{4176}(1565,·)$, $\chi_{4176}(869,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{29} a^{2}$, $\frac{1}{29} a^{3}$, $\frac{1}{1682} a^{4}$, $\frac{1}{1682} a^{5}$, $\frac{1}{146334} a^{6}$, $\frac{1}{146334} a^{7}$, $\frac{1}{8487372} a^{8}$, $\frac{1}{8487372} a^{9}$, $\frac{1}{4184274396} a^{10} - \frac{7}{144285324} a^{8} + \frac{1}{829226} a^{6} - \frac{7}{28594} a^{4} - \frac{7}{493} a^{2} - \frac{2}{17}$, $\frac{1}{4184274396} a^{11} - \frac{7}{144285324} a^{9} + \frac{1}{829226} a^{7} - \frac{7}{28594} a^{5} - \frac{7}{493} a^{3} - \frac{2}{17} a$

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

Class group and class number

$C_{2}\times C_{533234}$, which has order $1066468$ (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:  \( 481.70037561485367 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.15501312.6, 6.6.3359232.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$3$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$29$29.12.6.2$x^{12} - 20511149 x^{2} + 1784469963$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$