You might try the L-Series and Modular Forms Database (LMFDB) at

http://www.lmfdb.org/

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.