Go to https://sweet.ua.pt/apacetti/cnt-frames.html and click on "jac_cubic" on the right hand panel. It gives a link to download a pari-gp script that computes the Weierstrass equation of the jacobian of a general cubic and, from that, the discriminant, if you want. Artin, Tate and  Villegas have a paper explaining the theory too. It's on Villegas's webpage too.