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Properties

Label 12.0.29870126254...096.10
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{6}\cdot 11^{6}\cdot 13^{10}$
Root discriminant $194.81$
Ramified primes $2, 3, 11, 13$
Class number $1304448$ (GRH)
Class group $[4, 4, 81528]$ (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![122655625, 0, 20639500, 0, 1655491, 0, 80840, 0, 2683, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 60*x^10 + 2683*x^8 + 80840*x^6 + 1655491*x^4 + 20639500*x^2 + 122655625)
 
gp: K = bnfinit(x^12 + 60*x^10 + 2683*x^8 + 80840*x^6 + 1655491*x^4 + 20639500*x^2 + 122655625, 1)
 

Normalized defining polynomial

\( x^{12} + 60 x^{10} + 2683 x^{8} + 80840 x^{6} + 1655491 x^{4} + 20639500 x^{2} + 122655625 \)

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:  \(2987012625417735876007428096=2^{24}\cdot 3^{6}\cdot 11^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.81$
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:  $2, 3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3432=2^{3}\cdot 3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3432}(1,·)$, $\chi_{3432}(131,·)$, $\chi_{3432}(3301,·)$, $\chi_{3432}(3431,·)$, $\chi_{3432}(1453,·)$, $\chi_{3432}(1583,·)$, $\chi_{3432}(529,·)$, $\chi_{3432}(659,·)$, $\chi_{3432}(2773,·)$, $\chi_{3432}(2903,·)$, $\chi_{3432}(1849,·)$, $\chi_{3432}(1979,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{20} a^{5} + \frac{9}{20} a$, $\frac{1}{20} a^{6} + \frac{9}{20} a^{2}$, $\frac{1}{20} a^{7} + \frac{9}{20} a^{3}$, $\frac{1}{80} a^{8} - \frac{3}{40} a^{4} + \frac{5}{16}$, $\frac{1}{80} a^{9} - \frac{1}{40} a^{5} - \frac{19}{80} a$, $\frac{1}{286027811600} a^{10} + \frac{16324585}{2860278116} a^{8} + \frac{1984042089}{143013905800} a^{6} - \frac{21993493}{715069529} a^{4} + \frac{83408971921}{286027811600} a^{2} - \frac{325939045}{2860278116}$, $\frac{1}{633551602694000} a^{11} + \frac{45375872027}{126710320538800} a^{9} - \frac{7391834887771}{316775801347000} a^{7} + \frac{1508632035999}{63355160269400} a^{5} + \frac{241471788187}{1430139058000} a^{3} - \frac{7996496159929}{25342064107760} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{81528}$, which has order $1304448$ (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:  \( 4543.270357084286 \) (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

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{-429}) \), 3.3.169.1, \(\Q(\sqrt{26}, \sqrt{-66})\), 6.6.190102016.1, 6.0.525515088384.3, 6.0.853962018624.2

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$