Yes, finding points on elliptic curves is a deep subject; it's just about practical for smallish curves, but the algorithms are real pigs to implement and you need lots of them. If you can afford $400, buy a copy of the Magma computer algebra system; for small examples you can use

http://magma.maths.usyd.edu.au/calc/
and say for example

E:=EllipticCurve([0,0,1,-7,6]);

Rank(E);

IntegralPoints(E);

I can't quite remember the sequence of manipulations required to convert a quartic polynomial and a single rational point into an EllipticCurve and a map back to the y^2=quartic model.