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Properties

Label 12.0.29350398617...6001.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $90.29$
Ramified primes $3, 7, 17$
Class number $12285$ (GRH)
Class group $[3, 3, 1365]$ (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![18432, -117504, 243648, -74832, 63324, -23556, 5473, -1902, 267, 72, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 9*x^10 + 72*x^9 + 267*x^8 - 1902*x^7 + 5473*x^6 - 23556*x^5 + 63324*x^4 - 74832*x^3 + 243648*x^2 - 117504*x + 18432)
 
gp: K = bnfinit(x^12 - 6*x^11 - 9*x^10 + 72*x^9 + 267*x^8 - 1902*x^7 + 5473*x^6 - 23556*x^5 + 63324*x^4 - 74832*x^3 + 243648*x^2 - 117504*x + 18432, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} - 9 x^{10} + 72 x^{9} + 267 x^{8} - 1902 x^{7} + 5473 x^{6} - 23556 x^{5} + 63324 x^{4} - 74832 x^{3} + 243648 x^{2} - 117504 x + 18432 \)

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:  \(293503986178850203226001=3^{16}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.29$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1071}(1,·)$, $\chi_{1071}(934,·)$, $\chi_{1071}(103,·)$, $\chi_{1071}(970,·)$, $\chi_{1071}(781,·)$, $\chi_{1071}(562,·)$, $\chi_{1071}(307,·)$, $\chi_{1071}(52,·)$, $\chi_{1071}(373,·)$, $\chi_{1071}(118,·)$, $\chi_{1071}(985,·)$, $\chi_{1071}(883,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} + \frac{1}{64} a^{6} + \frac{3}{64} a^{5} + \frac{3}{64} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{256} a^{8} - \frac{1}{128} a^{7} - \frac{1}{64} a^{6} - \frac{5}{128} a^{5} + \frac{15}{256} a^{4} + \frac{11}{64} a^{3} - \frac{15}{64} a^{2} + \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{3072} a^{9} - \frac{1}{512} a^{8} - \frac{1}{128} a^{7} + \frac{15}{512} a^{6} + \frac{17}{1024} a^{5} - \frac{7}{256} a^{4} - \frac{155}{768} a^{3} + \frac{1}{64} a^{2} - \frac{1}{8} a$, $\frac{1}{6144} a^{10} - \frac{1}{6144} a^{9} + \frac{1}{1024} a^{8} - \frac{1}{1024} a^{7} + \frac{7}{2048} a^{6} - \frac{127}{2048} a^{5} + \frac{73}{1536} a^{4} + \frac{329}{1536} a^{3} - \frac{13}{64} a^{2} - \frac{7}{32} a - \frac{1}{4}$, $\frac{1}{783334834176} a^{11} + \frac{8734699}{391667417088} a^{10} + \frac{12722901}{261111611392} a^{9} + \frac{13902049}{65277902848} a^{8} + \frac{1025656713}{261111611392} a^{7} - \frac{324863103}{130555805696} a^{6} + \frac{19976011561}{783334834176} a^{5} + \frac{1370097355}{97916854272} a^{4} + \frac{11148950197}{65277902848} a^{3} + \frac{881123445}{4079868928} a^{2} + \frac{1098508289}{4079868928} a + \frac{102185571}{509983616}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{1365}$, which has order $12285$ (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:  \( 1065625.9379812907 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-119}) \), 3.3.3969.1, \(\Q(\sqrt{-7}, \sqrt{17})\), 6.0.110270727.2, 6.6.77394297393.1, 6.0.541760081751.3

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$