You might try the L-Series and Modular Forms Database (LMFDB) at
In particular, you can look up elliptic curves using Weierstrass equations at
http://www.lmfdb.org/EllipticCurve/Q/
Addendum: For the particular curve that you listed, entering $[0,-39,0,320,1024]$into the LMFDB yields the following information:
Minimal Weierstrass equation$$y^2+xy+y=x^3−x^2−12x+15.$$Mordell-Weil group structure $E(\mathbb Q)=\mathbb Z$.
Infinite order Mordell-Weil generator $P=(1,1)$.
Integral points$$\{(−3,5), (−1,5), (1,1), (3,−1), (5,5), (13,37), (25,109)\}.$$Note: only one of each pair $\pm P$ is listed.