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Properties

Label 12.0.59779054097...5641.4
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{10}\cdot 29^{6}$
Root discriminant $79.08$
Ramified primes $3, 13, 29$
Class number $15768$ (GRH)
Class group $[2, 6, 1314]$ (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![173113225, -24466635, 39465521, -4854123, 3923740, -404221, 217076, -17686, 7040, -408, 127, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 127*x^10 - 408*x^9 + 7040*x^8 - 17686*x^7 + 217076*x^6 - 404221*x^5 + 3923740*x^4 - 4854123*x^3 + 39465521*x^2 - 24466635*x + 173113225)
 
gp: K = bnfinit(x^12 - 4*x^11 + 127*x^10 - 408*x^9 + 7040*x^8 - 17686*x^7 + 217076*x^6 - 404221*x^5 + 3923740*x^4 - 4854123*x^3 + 39465521*x^2 - 24466635*x + 173113225, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 127 x^{10} - 408 x^{9} + 7040 x^{8} - 17686 x^{7} + 217076 x^{6} - 404221 x^{5} + 3923740 x^{4} - 4854123 x^{3} + 39465521 x^{2} - 24466635 x + 173113225 \)

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:  \(59779054097312062075641=3^{6}\cdot 13^{10}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.08$
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:  $3, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1131=3\cdot 13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1131}(608,·)$, $\chi_{1131}(1,·)$, $\chi_{1131}(610,·)$, $\chi_{1131}(521,·)$, $\chi_{1131}(1130,·)$, $\chi_{1131}(523,·)$, $\chi_{1131}(173,·)$, $\chi_{1131}(784,·)$, $\chi_{1131}(1043,·)$, $\chi_{1131}(88,·)$, $\chi_{1131}(347,·)$, $\chi_{1131}(958,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a$, $\frac{1}{50593243999287449491976254565} a^{11} + \frac{2205718887966755446913476176}{50593243999287449491976254565} a^{10} + \frac{18780597634985704744697387522}{50593243999287449491976254565} a^{9} + \frac{11840513373226966061628495807}{50593243999287449491976254565} a^{8} + \frac{2829204840536171276956026980}{10118648799857489898395250913} a^{7} + \frac{11496666294264361570237978239}{50593243999287449491976254565} a^{6} + \frac{16285210974111333737333206701}{50593243999287449491976254565} a^{5} - \frac{15370932346019099372331573761}{50593243999287449491976254565} a^{4} - \frac{4968583947265032521739647251}{10118648799857489898395250913} a^{3} + \frac{20000609631275263899863164912}{50593243999287449491976254565} a^{2} + \frac{16044292150662249195127666226}{50593243999287449491976254565} a + \frac{2840251857933445372847214159}{10118648799857489898395250913}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{1314}$, which has order $15768$ (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:  \( 120.78403136265631 \) (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

\(\Q(\sqrt{-1131}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-87}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-87})\), 6.0.244497554379.6, \(\Q(\zeta_{13})^+\), 6.0.18807504183.8

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$