Why do you want to decrease the coefficients? If it is to search for points, then possibly what you're looking for is the theory of homogeneous spaces. Thus an equation of the form $C:y^2=ax^4+\cdots+e$, even if it does not have a rational point, represents an element of order $2$ in the Weil-Chatelet group of its Jacobean, which is an elliptic curve $E$, and there is a map $C\to E$ defined over your base field. Conversely, starting with an elliptic curve $E$, there exist homogeneous spaces of various orders $m$, and, to some extent, the coefficients of these homogeneous spaces will be smaller than the coefficients of the original curve. If $m$ is not too large, say $2\le m\le 6$, it's probably feasible to write down (some of) the associated homogeneous spaces. Note that rational points on the homogeneous spaces will then map to rational points on $E$.
This is closely related to performing an $m$-descent on $E$. The first place I'd suggest looking is Cremona's book , which in addition to a lot of tables (since expanded and transferred online) includes a very nice description of practically constructing homogeneous spaces and indicates why their coefficients are often smaller than those on the original curve. See in particular Chapter 3.
 MR1628193 Cremona, J. E. Algorithms for modular elliptic curves. Second edition. Cambridge University Press, Cambridge, 1997. Freely available online at http://homepages.warwick.ac.uk/~masgaj/book/fulltext/index.html