WARNING: Once the Magma integral points code was fixed, the Magma developers found bugs in the Sage integral_points code, as of Sage-4.6.1. See trac.sagemath.org/sage_trac/ticket/10152

It's like this: "y^2 + a1*xy + a3*y = x^3 + a2*x^2 + a4*x + a6". You can get this out of Sage as follows: "var('a1,a2,a3,a4,a6'); EllipticCurve([a1,a2,a3,a4,a6])" The way to *remember this is to think of x as degree 2, y as degree 3, and a_i as degree i; then the Weierstrass equation is homogenous of degree 6.

What is your application? E.g., are you interested in whether or not an abelian variety is absolutely simple? I've thought about this problem in the context of endomorphism rings of modular abelian varieties; there one can tell whether or not the algebra is a matrix algebra using results of Ribet and Lario (?), but finding an explicit representation as a matrix algebra is harder.

@Dror: Yes, C.rational_points(bound=1000) would give the same result; however it is stupid generic Python code and is literally almost 100,000 times slower than the little Cython function I wrote, so it would take 2 weeks instead of 13 seconds. I don't know what you mean by "This should really be a simple exercise in either 7-adics or the number field generated by the polynomial on the left..." The curve does have rational points (the two at infinity), so I don't see how to rule out rational points by working p-adically. Maybe you can rule out integral points, but that is different.

In my experience with graduate admissions (Harvard, University of Washington) the math subject GRE score is important---certainly much more so than other comments above suggest. For example, the Berkeley math grad application page says "Experience has shown that the score on the Mathematics Subject GRE is a partial indicator of preparation for Berkeley's PhD program. A score below the 80th percentile suggests inadequate preparation and must be balanced by other evidence if a favorable admission decision is to be reached." See math.berkeley.edu/graduate_phd.html