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Properties

Label 12.0.9291682743890625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 13^{8}$
Root discriminant $21.41$
Ramified primes $3, 5, 13$
Class number $4$
Class group $[2, 2]$
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![25, 50, 195, -60, 431, 12, 377, -128, 138, -14, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 13*x^10 - 14*x^9 + 138*x^8 - 128*x^7 + 377*x^6 + 12*x^5 + 431*x^4 - 60*x^3 + 195*x^2 + 50*x + 25)
 
gp: K = bnfinit(x^12 - x^11 + 13*x^10 - 14*x^9 + 138*x^8 - 128*x^7 + 377*x^6 + 12*x^5 + 431*x^4 - 60*x^3 + 195*x^2 + 50*x + 25, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 13 x^{10} - 14 x^{9} + 138 x^{8} - 128 x^{7} + 377 x^{6} + 12 x^{5} + 431 x^{4} - 60 x^{3} + 195 x^{2} + 50 x + 25 \)

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:  \(9291682743890625=3^{6}\cdot 5^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.41$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(1,·)$, $\chi_{195}(131,·)$, $\chi_{195}(74,·)$, $\chi_{195}(139,·)$, $\chi_{195}(29,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(16,·)$, $\chi_{195}(146,·)$, $\chi_{195}(61,·)$, $\chi_{195}(94,·)$, $\chi_{195}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{755} a^{10} - \frac{346}{755} a^{9} + \frac{358}{755} a^{8} - \frac{39}{755} a^{7} - \frac{257}{755} a^{6} - \frac{318}{755} a^{5} - \frac{298}{755} a^{4} - \frac{323}{755} a^{3} - \frac{324}{755} a^{2} - \frac{60}{151} a - \frac{57}{151}$, $\frac{1}{6058308628445} a^{11} + \frac{2305274479}{6058308628445} a^{10} - \frac{2134501577017}{6058308628445} a^{9} - \frac{1938778936554}{6058308628445} a^{8} + \frac{2367682718933}{6058308628445} a^{7} - \frac{2073626969308}{6058308628445} a^{6} - \frac{664617231333}{6058308628445} a^{5} + \frac{29065935497}{195429310595} a^{4} + \frac{413432499561}{6058308628445} a^{3} + \frac{347793044225}{1211661725689} a^{2} - \frac{88294744774}{1211661725689} a + \frac{524375881260}{1211661725689}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( -\frac{529617992}{40121249195} a^{11} + \frac{668517648}{40121249195} a^{10} - \frac{6889267202}{40121249195} a^{9} + \frac{9101711556}{40121249195} a^{8} - \frac{14680127089}{8024249839} a^{7} + \frac{85456721409}{40121249195} a^{6} - \frac{200310704977}{40121249195} a^{5} + \frac{1064846883}{1294233845} a^{4} - \frac{38208275105}{8024249839} a^{3} + \frac{121559644291}{40121249195} a^{2} - \frac{15755410925}{8024249839} a + \frac{4215357579}{8024249839} \) (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:  \( 615.54450504 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.771147.1, 6.6.3570125.1, 6.0.96393375.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$