Is it impossible to describe Jacobian of hyperelliptic curves by an explicit equation?

Question

Let $C$ be a hyperelliptic curve and $J(C)$ be its Jacobian. $J(C)$ is abelian variety, especially, projective variety, so it should have equation which defines $J(C)$.

$J(C)$ should be defined by equations, but it is almost impossible to describe its equation by hand?

(For example, Let $C$ be hyperelliptic curve given by equation $y^{19}=x^2(1-x)$. What is an explicit equation of $J(C)$ ?)

Answer 1

Yes, $J(C)$ can be described in a very explicit way, see Section 4 in [M. A. Reid, The complete intersection of two or more quadrics].