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Properties

Label 12.0.4469547301936929.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 19^{10}$
Root discriminant $20.15$
Ramified primes $3, 19$
Class number $3$
Class group $[3]$
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![49, -35, 32, -107, 27, 5, 75, 9, 13, -18, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - x^10 - 18*x^9 + 13*x^8 + 9*x^7 + 75*x^6 + 5*x^5 + 27*x^4 - 107*x^3 + 32*x^2 - 35*x + 49)
 
gp: K = bnfinit(x^12 - x^11 - x^10 - 18*x^9 + 13*x^8 + 9*x^7 + 75*x^6 + 5*x^5 + 27*x^4 - 107*x^3 + 32*x^2 - 35*x + 49, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - x^{10} - 18 x^{9} + 13 x^{8} + 9 x^{7} + 75 x^{6} + 5 x^{5} + 27 x^{4} - 107 x^{3} + 32 x^{2} - 35 x + 49 \)

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:  \(4469547301936929=3^{6}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(57=3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(37,·)$, $\chi_{57}(7,·)$, $\chi_{57}(8,·)$, $\chi_{57}(11,·)$, $\chi_{57}(46,·)$, $\chi_{57}(49,·)$, $\chi_{57}(50,·)$, $\chi_{57}(20,·)$, $\chi_{57}(56,·)$, $\chi_{57}(26,·)$, $\chi_{57}(31,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{77} a^{10} + \frac{2}{77} a^{9} - \frac{4}{77} a^{8} - \frac{4}{77} a^{7} + \frac{4}{77} a^{6} - \frac{9}{77} a^{5} - \frac{3}{11} a^{4} + \frac{1}{77} a^{3} - \frac{1}{77} a^{2} + \frac{24}{77} a + \frac{2}{11}$, $\frac{1}{87857} a^{11} - \frac{9}{1793} a^{10} - \frac{3354}{87857} a^{9} + \frac{1570}{87857} a^{8} - \frac{1623}{87857} a^{7} - \frac{2419}{87857} a^{6} + \frac{11292}{87857} a^{5} - \frac{29086}{87857} a^{4} - \frac{13281}{87857} a^{3} - \frac{39463}{87857} a^{2} + \frac{328}{12551} a - \frac{499}{1793}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{179}{7987} a^{11} - \frac{30}{1141} a^{10} - \frac{200}{7987} a^{9} - \frac{3079}{7987} a^{8} + \frac{2720}{7987} a^{7} + \frac{1720}{7987} a^{6} + \frac{10826}{7987} a^{5} + \frac{1130}{7987} a^{4} + \frac{6250}{7987} a^{3} - \frac{9074}{7987} a^{2} + \frac{1010}{1141} a + \frac{3}{163} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1924.35460113 \)
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

\(\Q(\sqrt{57}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-19}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{-19})\), 6.6.66854673.1, 6.0.3518667.1, 6.0.2476099.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$