Class ReductionData

                      object --+    
                               |    
structure.sage_object.SageObject --+
                                   |
                                  ReductionData

Reduction data for a genus 2 curve.

How to read \code{local_data} attribute, i.e., if this class is R,
then the following is the meaning of \code{R.local_data[p]}.

    For each prime number $p$ dividing the discriminant of $y^2+Q(x)y=P(x)$,
    there are two lines.

    The first line contains information about the stable reduction
    after field extension. Here are the meanings of the symbols of
    stable reduction :

    (I)   The stable reduction is smooth (i.e. the curve has potentially 
          good reduction).
          
    (II)  The stable reduction is an elliptic curve $E$ with an ordinary double 
          point. $j$ mod $p$ is the modular invariant of $E$.
          
    (III) The stable reduction is a projective line with two ordinary double 
          points.
          
    (IV)  The stable reduction is two projective lines crossing transversally
          at three points.
          
    (V)   The stable reduction is the union of two elliptic curves $E_1$ and $E_2$
          intersecting transversally at one point. Let $j_1$, $j_2$ be their modular
          invariants, then $j_1+j_2$ and $j_1 j_2$ are computed (they are numbers
          mod $p$).
          
    (VI)  The stable reduction is the union of an elliptic curve $E$ and a 
          projective line which has an ordinary double point. These two 
          components intersect transversally at one point. $j$ mod $p$ is the 
          modular invariant of $E$.
          
    (VII) The stable reduction is as above, but the two components are both 
          singular.

    In the cases (I) and (V), the Jacobian $J(C)$ has potentially good
    reduction.  In the cases (III), (IV) and (VII), $J(C)$ has
    potentially multiplicative reduction. In the two remaining cases,
    the (potential) semi-abelian reduction of $J(C)$ is extension of an
    elliptic curve (with modular invariant $j$ mod $p$) by a torus.

    The second line contains three data concerning the reduction at $p$
    without any field extension.

    \begin{enumerate}
      \item The first symbol describes the REDUCTION AT $p$ of $C$. We use
        the symbols of Namikawa-Ueno for the type of the reduction
        (Namikawa, Ueno:"The complete classification of fibers in
        pencils of curves of genus two", Manuscripta Math.,
        vol. 9, (1973), pages 143-186.) The reduction symbol is
        followed by the corresponding page number (or just an
        indiction) in the above article.  The lower index is
        printed by { }, for instance, [I{2}-II-5] means
        [I_2-II-5].  Note that if $K$ and $K'$ are Kodaira symbols for
        singular fibers of elliptic curves, [K-K'-m] and [K'-K-m]
        are the same type. Finally, [K-K'--1] (not the same as
        [K-K'-1]) is [K'-K-alpha] in the notation of
        Namikawa-Ueno. The figure [2I_0-m] in Namikawa-Ueno, page
        159 must be denoted by [2I_0-(m+1)].

      \item The second datum is the GROUP OF CONNECTED COMPONENTS
        (over an ALGEBRAIC CLOSURE (!) of $\F_p$) of the Neron model of
        J(C). The symbol (n) means the cyclic group with n
        elements. When n=0, (0) is the trivial group (1). \code{H{n}} is
        isomorphic to (2)x(2) if n is even and to (4) otherwise.

        Note -- The set of rational points of $\Phi$ can be
        computed using Theorem 1.17 in S. Bosch and Q. Liu
        "Rational points of the group of components of a
        N\'eron model", Manuscripta Math. 98 (1999), 275-293. 

      \item Finally, $f$ is the exponent of the conductor of $J(C)$ at $p$.
      
    \end{enumerate}

    \note{Warning -- Be careful regarding the formula:
    $$
     \text{valuation of the naive minimal discriminant}
      = f + n - 1 + 11c(X).
    $$
    (Q. Liu : "Conducteur et discriminant minimal de courbes de 
    genre 2", Compositio Math. 94 (1994) 51-79, Theoreme 2)
    is valid only if the residual field is algebraically 
    closed as stated in the paper. So this equality does not
    hold in general over $\Q_p$. The fact is that the minimal 
    discriminant may change after unramified extension. 
    One can show however that, at worst, the change will
    stabilize after a quadratic unramified extension
    (Q. Liu : "Modeles entiers de courbes hyperelliptiques sur un
    corps de valuation discrete", Trans. AMS 348 (1996), 4577-4610,
    \S 7.2, Proposition 4).
    }

x.__init__(...) initializes x; see x.__class__.__doc__ for signature