Go to this web site 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.
Felipe Voloch
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