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I always use PARI when I need to do computations and I am a big fan of this little program. I believe that it is possible to do in PARI everything you can do with such big programs as Maple and Mathematica. Well… almost everything. Here I’d like to present some tricks to do things in PARI that seem impossible from first sight, or just convenient hints. Readers are very welcome to publish their own tricks in comments. This way we may create something like a library of tricks.

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It seems when people talk about modular forms they tend to forget that they are very related to families of elliptic curves. Here I want to explain some simple way to understand the connection. We will consider modular forms for the full modular group $SL(2, \mathbf Z)$.

So consider the simplest family of elliptic curves, the Weierstrass family:

$y^2=x^3+ax+b.$

Recently I was trying to understand what’s behind the mysterious condition

$\sum_i [z_i]\wedge[1-z_i] = 0\in \Lambda^2 \mathbb C^\times$

for elements $z_i\in \mathbb C$ ($z_i\neq 0, 1$) to define an element $\sum_i[z_i]$ in the Bloch group. It appears that the condition naturally appears if one studies hyperbolic $3$-manifolds.