TITLE: Birational Equivalence of Diophantine Equations and Elliptic Curves
QUESTION [10 upvotes]: A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given for this problem describes a method for determining solutions to diophantine equations by finding a "birational equivalence" to the solution set of an elliptic curve. 
The result I am seeking depends on the solution set to a quartic diophantine equation of two variables and I believe that there is only one solution and no more (which was indicated by a program which checked possible values up to a high number).
So my question is, is there a general method for determining such a birational equivalence between solutions sets of diophantine equations and elliptic curves? And where is a good source for an undergraduate to learn to use these tools?

REPLY [5 votes]: (This is a long comment re MacLeod's answer.)
We can also combine the two cases together. Assume a quartic polynomial to be made a square,
$$pu^4+qu^3+ru^2+su+t=z_1^2\tag1$$
has a known rational point, call it $w$. We substitute $u=v+w$ and collect the new variable $v$, 
$$c_4v^4+c_3v^3+c_2v^2+c_1v+\color{blue}{c_0^2}=z_2^2\tag2$$
where the $c_i$ are polynomials in $w$ and the coefficients of $(1)$. The constant term of $(2)$ turns out to be a square, specifically, $c_0^2:=pw^4+qw^3+rw^2+sw+t$.
Let $v=1/x,\,$ $z_2=c_0\, z_3/x^2$ and $(2)$ becomes,
$$x^4+d_3x^3+d_2x^2+d_1x+d_0=z_3^2\tag3$$
which is the Case 1 and then be solved as explained by MacLeod.