I would like to automate a huge amount of computation that involve basic arithmetic operations with $p$-adic numbers. I have found a Mathematica packages for it, but it is old and acts quite erratically. Do you know of any computational software that does it reliably?

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Could you be more explicit about just which operations you want to do, and where (as in, what $p$-adic fields)? And could you say more fully what you mean by "automate"? –  Lubin Jun 30 '11 at 17:18

There is p-adic arithmetic in the free software programs SAGE, PARI and GAP, but their main limitation (i don't know if things have changed recently) was their inability to with relative extension, that is extensions of another field which is itself a proper extension of $\mathbb{Q}_p$. However if you have limited need for extensions they (mostly SAGE and PARI, because GAP is more group-theory oriented) have a very good interface to work with p-adic numbers.