  Let $A$ be an abelian variety of dimension $g$ together with a principal
polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$
be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup
of $A[\ell]$ which is maximal isotropic for the Riemann form associated to
$\phi$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient
abelian variety together with a polarization compatible with $\phi$. Then $B$,
as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also
defined over $k$. In this paper, we describe an algorithm that takes as input a
theta null point of $A$ and a polynomial system defining $K$ and outputs a
theta null point of $B$ as well as formulas for the isogeny $f$. We obtain a
complexity of $\tilde{O}(\ell^{\frac{rg}{2}})$ operations in $k$ where $r=2$
(resp. $r=4$) if $\ell$ is a sum of two squares (resp. if $\ell$ is a sum of
four squares) which constitutes an improvement over the algorithm described in
[7]. We note that the algorithm is quasi-optimal if $\ell$ is a sum of two
squares since its complexity is quasi-linear in the degree of $f$.
