  We present an efficient endomorphism for the Jacobian of a curve $C$ of genus
2 (hyperelliptic) for divisors having a Non disjoint support. This extends the
work of Costello and Lauter in [12] who calculated explicit formulae for
divisor doubling and addition of divisors with disjoint support in
$\mathbb{J}(C)$ using only base field operations. Explicit formulae is
presented for this third case and a slightly different approach for divisor
doubling.
