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Properties

Label 12.0.90919078229...5264.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 47^{6}$
Root discriminant $120.20$
Ramified primes $2, 3, 7, 47$
Class number $207648$ (GRH)
Class group $[2, 4, 25956]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10779215329, 0, 3669520112, 0, 292780860, 0, 8098194, 0, 97196, 0, 517, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 517*x^10 + 97196*x^8 + 8098194*x^6 + 292780860*x^4 + 3669520112*x^2 + 10779215329)
 
gp: K = bnfinit(x^12 + 517*x^10 + 97196*x^8 + 8098194*x^6 + 292780860*x^4 + 3669520112*x^2 + 10779215329, 1)
 

Normalized defining polynomial

\( x^{12} + 517 x^{10} + 97196 x^{8} + 8098194 x^{6} + 292780860 x^{4} + 3669520112 x^{2} + 10779215329 \)

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:  \(9091907822989955933835264=2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.20$
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:  $2, 3, 7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3948=2^{2}\cdot 3\cdot 7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{3948}(1,·)$, $\chi_{3948}(1315,·)$, $\chi_{3948}(1691,·)$, $\chi_{3948}(1129,·)$, $\chi_{3948}(1879,·)$, $\chi_{3948}(941,·)$, $\chi_{3948}(2255,·)$, $\chi_{3948}(3383,·)$, $\chi_{3948}(3385,·)$, $\chi_{3948}(187,·)$, $\chi_{3948}(3197,·)$, $\chi_{3948}(377,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{47} a^{2}$, $\frac{1}{47} a^{3}$, $\frac{1}{2209} a^{4}$, $\frac{1}{2209} a^{5}$, $\frac{1}{103823} a^{6}$, $\frac{1}{103823} a^{7}$, $\frac{1}{4879681} a^{8}$, $\frac{1}{4879681} a^{9}$, $\frac{1}{229345007} a^{10}$, $\frac{1}{229345007} a^{11}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{25956}$, which has order $207648$ (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:  \( \frac{1}{229345007} a^{10} + \frac{10}{4879681} a^{8} + \frac{35}{103823} a^{6} + \frac{51}{2209} a^{4} + \frac{29}{47} a^{2} + 4 \),  \( \frac{1}{103823} a^{6} + \frac{6}{2209} a^{4} + \frac{9}{47} a^{2} + 2 \),  \( \frac{1}{2209} a^{4} + \frac{4}{47} a^{2} + 1 \),  \( \frac{1}{47} a^{2} + 1 \),  \( \frac{1}{103823} a^{6} + \frac{6}{2209} a^{4} + \frac{8}{47} a^{2} + 1 \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140.7987960054707 \) (assuming GRH)
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

\(\Q(\sqrt{-141}) \), \(\Q(\sqrt{-329}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-141})\), 6.0.430754151744.3, 6.0.111677002304.6, \(\Q(\zeta_{21})^+\)

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$