  We describe the minimal number of critical points and the minimal number $s$
of singular fibres for a non isotrivial fibration of a surface $S$ over a curve
$B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular
fibre with $4$ nodes.
  Then we consider the associated factorizations in the mapping class group and
in the symplectic group. We describe explicitly which products of transvections
on homologically independent and disjoint circles are a commutator in the
Symplectic group $Sp (2g, \mathbb{Z})$.
