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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 singularfibre with $4$ nodes.  Then we consider the associated factorizations in the mapping class group andin the symplectic group. We describe explicitly which products of transvectionson homologically independent and disjoint circles are a commutator in theSymplectic group $Sp (2g, \mathbb{Z})$.

Fibred algebraic surfaces and commutators in the Symplectic group