As part of a physics project studying renormalization group flows of scalar field theories, I've come across the following quartic plane curve in the variables $m$ and $n$: $$ 36 + 16 m^4 - 108 n + 105 n^2 - 18 n^3 + n^4 - 8 m^3 (-18 + 17 n) + m^2 (420 - 468 n + 81 n^2) + m (216 - 408 n + 234 n^2 - 34 n^3)=0. $$ I would like to know the list of integer solutions to this equation.
Searching all values of $m$ and $n$ up to a few thousand, I've found only (-2,-2), (-1,2), and (1,4). I know the curve has two nodal points, one at (-1,2) and another at (-1/5, 2/5) and thus should have genus one. Playing around with conics and lines through nodal points, I can generate some further rational solutions, for example (-88857223/103456502,-73582856/51728251), but my interest is in whether there exist further integer solutions.
Using the Maple algcurves package, I can put this curve in Weierstrass form, but the coefficients involve roots of $z^2 - 10 z + 7$, and so I'm not sure that's actually a step forward.
Is there some software I could use to solve this Diophantine equation? Is there an efficient pen and paper method?
Update: I made some progress with the Maple algcurves package. By calling the Weierstrass form command with the option [-2, -2, 1], I find the cubic $$ y^2 = x^3 + 680244480 x + 66648721563648 $$
Through google search and talking with some number theorist friends, I came across the following two papers by Tzanakis and Stroeker: Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms and Computing all integer solutions of a genus 1 equation.
As often happens, my problem is simpler than the general case. As pointed out by Viktor Vaughn in the comments, the curve has rank one. Thus there is no need for this Lenstra, Lenstra, Lovasz algorithm toward the end of the demonstration. The lattice in question is only two dimensional, and Gauss's old algorithm suffices.
At any rate, if I have understood the procedure (a big IF in my case), I need to search through only a handful of points to demonstrate that I have in fact found all the integer solutions. The lattice of rational points can be written in the form $$ P = s T + q P_1 $$ where $s \in \{0, 1 \}$ and $q \in {\mathbb Z}$.
(According to Sage, I can take $T = (-34992, 0)$ and $P_1 = ( -11664, 7558272)$ in the $(x,y)$ coordinates. Note $T$ is the point (1,4) in the original $(m,n)$ coordinates while $P_1$ is (8/5, 7/10). The other $(m,n)$-integer points $(-2,-2)$ and $(-1,2)$ have been pushed off to infinity by the birational transformation.)
The Tzanakis-Stroeker procedure tells me it is enough to check all points with $|q| \leq 5$, IF I followed the instructions correctly.
In conclusion, (-2,-2), (-1,2), and (1,4) are indeed the only integer solutions.
Comments welcome, especially if I've said something obviously wrong. I'd like to make sure I've done things correctly.