(i) Some form of symbolic algebra package would be a great help for the algebraic manipulation, especially the transformation of quartics to elliptic curves. I use the MSDOS version of Derive, which is really old but works fine. Sometimes it has problems factorising some complicated expressions, so I then use Mathematica.

(ii) The numbers involved in the calculation can be very large integers. Certainly you will very quickly outstrip the ability of normal programming languages to store integers. There is thus a requirement for some form of multiple precision software. I use UBASIC , developed by H.Kida, which is a wonderfully powerful yet simple package. It is also FREE. There are limitations on the complexity you can build into a UBASIC code, so something bigger might be useful. I also use PARI-GP , but you probably need slightly more mathematical knowledge to make effective use of PARI.

(iii) Anybody dealing with elliptic curves MUST get a copy of John Cremona's mwrank program for the system they use. This is an extremely powerful program and is really the de facto standard piece of software for elliptic curves.