TITLE: Simplest complex curves with isomorphic Jacobian
QUESTION [5 upvotes]: Hi. 
What are the simplest examples to have in mind of non-isomorphic smooth (complex) algebraic curves with isomorphic jacobian variety?

REPLY [3 votes]: Slightly tangential -- the following paper of Bjorn Poonen offers some interesting computational perspectives on curves of genus at least $2$ which is as Francesco mentions, the lowest possible genus where such examples exist. This seems like it might be relevant.
http://www-math.mit.edu/~poonen/papers/ants2.pdf
I think the following is an interesting question to ask over a fixed number field $K$: Can one make tables of genus $2$ curves over $K$ ordered in some lexicographic fashion by the invariants? Perhaps the curves would be modelled by singular plane curves $y^2 = f(x)$ where $f$ has degree $5$ or $6$. The first pair $(X,Y)$ with isomorphic Jacobian variety to occur in this table might satisfy the request that the pair $(X,Y)$ be the "simplest" such pair to be defined over $K$. 
Poonen notes that not every not every hyperelliptic curve has a model of the form $y^2 = f(x)$ over
$k$ a field, because the quotient of the curve by its hyperelliptic involution might be a twist of $\mathbb{P}^1$ i.e. birational over $k$ to a conic in $\mathbb{P}^2$ without a $k$-rational point. However when $g$ is even, it seems to turn out (I'm not sure why) that the $k$-form of $\mathbb{P}^1$ has a $k$-point and the hyperelliptic curve has a model $y^2 = f(x)$, so for $g = 2$ we should be okay.