Let $f: \mathbb{C}^g/L\to\mathbb{C}^{g'}/L'$ be an isogeny of complex tori, i.e. a surjective Lie group morphism with finite kernel.
Is it obvious that $g\ge g'$ ?
It is easy to show that $f$ is induced by a linear map $\mathbb{C}^g\to\mathbb{C}^{g'}$ that is injective, but I cannot see why this map should be an isomorphism. A surjective morphism of Lie group is not necessarily a submersion, right ?
In general, any Lie group homomorphism whose kernel is a discrete subgroup of the center is a normal covering. In particular it is a local isomorphism so in your case $g = g'$. For a proof, see proposition 1.19 of http://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf.