  We give a practical method to compute the 2-torsion subgroup of the Jacobian
of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on
the correspondence between the 2-torsion subgroup and the theta hyperplanes to
the curve. The correspondence is used to explicitly write down a
zero-dimensional scheme whose points correspond to elements of the $2$-torsion
subgroup. Using $p$-adic or complex approximations (obtained via Hensel lifting
or homotopy continuation and Newton-Raphson) and lattice reduction we are then
able to determine the points of our zero-dimensional scheme and hence the
$2$-torsion points. We demonstrate the practicality of our method by computing
the $2$-torsion of the modular Jacobians $J_{0}\left( N \right)$ for $N = 42,
55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg
conjecture for these values.
