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Properties

Label 12.0.27927616772...3376.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $41.74$
Ramified primes $2, 7, 17$
Class number $180$ (GRH)
Class group $[3, 60]$ (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![32761, 0, 26083, 0, 8625, 0, 1527, 0, 169, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 15*x^10 + 169*x^8 + 1527*x^6 + 8625*x^4 + 26083*x^2 + 32761)
 
gp: K = bnfinit(x^12 + 15*x^10 + 169*x^8 + 1527*x^6 + 8625*x^4 + 26083*x^2 + 32761, 1)
 

Normalized defining polynomial

\( x^{12} + 15 x^{10} + 169 x^{8} + 1527 x^{6} + 8625 x^{4} + 26083 x^{2} + 32761 \)

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:  \(27927616772217573376=2^{12}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.74$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(476=2^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{476}(1,·)$, $\chi_{476}(67,·)$, $\chi_{476}(101,·)$, $\chi_{476}(135,·)$, $\chi_{476}(137,·)$, $\chi_{476}(103,·)$, $\chi_{476}(205,·)$, $\chi_{476}(237,·)$, $\chi_{476}(307,·)$, $\chi_{476}(171,·)$, $\chi_{476}(407,·)$, $\chi_{476}(33,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{70366619} a^{10} + \frac{26332549}{70366619} a^{8} + \frac{15796818}{70366619} a^{6} + \frac{19491790}{70366619} a^{4} + \frac{30594685}{70366619} a^{2} - \frac{450931}{70366619}$, $\frac{1}{12736358039} a^{11} - \frac{1099533355}{12736358039} a^{9} - \frac{4558033417}{12736358039} a^{7} + \frac{3326722883}{12736358039} a^{5} + \frac{3689658873}{12736358039} a^{3} + \frac{3095680305}{12736358039} a$

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

Class group and class number

$C_{3}\times C_{60}$, which has order $180$ (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:  \( 246.505463083 \) (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

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-119}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{7}, \sqrt{-17})\), \(\Q(\zeta_{28})^+\), 6.0.754951232.2, 6.0.82572791.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$