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I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.

Question

Which computer algebra system is best for what, and what is the easiest/most fun(?) way to learn how to deal with them?

I should mention that I do (arguably) Arithmetic Geometry. So, ideally it would be I should invest my time in learning something that is capable of making the abstract existence theorems in that field explicit (see for example: http://mathoverflow.net/questions/52125/how-to-get-explicit-unramified-covers-of-an-elliptic-curve, or another example is normalizations).

P.S. I don't have access to Magma. How powerful is their "online calculator", and is it worthwhile to learn Magma just to use the online version?
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Roadmap to Computer Algebra Systems Usage for Algebraic Geometry

I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.

Question

Which computer algebra system is best for what, and what is the easiest/most fun(?) way to learn how to deal with them?

I should mention that I do (arguably) Arithmetic Geometry. So, ideally it would be something that is capable of making the abstract existence theorems in that field explicit (see for example: http://mathoverflow.net/questions/52125/how-to-get-explicit-unramified-covers-of-an-elliptic-curve, or another example is normalizations).

P.S. I don't have access to Magma. How powerful is their "online calculator", and is it worthwhile to learn Magma just to use the online version?