# Integral points in elliptic curves

I am getting a number of elliptic curves for which I need to find integral points. One of them is

$$y^2= x^3 - 39x^2 + 320x + 1024$$

(so I have Weierstrass coefficients). My question is, how to find them easily in Table 7 on http://johncremona.github.io/ecdata/ (or in some other location). For example, it seems that the above curve in not present in that table. Moreover, how to find all data (rank, torsion...) of a curve if I have its W. coeff.?

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.

• From: "Zoran Stanic" <zstanic@math.rs> Date: Fri, 06 Oct 2017 14:55:52 +0200 Dear Professor Silverman, Sorry for interrupting you directly, but unfortunately I am new on MathOverflow, so I cannot add a comment. About your answer to my question regarding Integral points in elliptic curves: Are you sure that ALL integral points are listed (up to reversing the sign) for that particular or any other curve? (In other words, would I be noticed if there is a possible existence of other int. points?) Oct 6, 2017 at 13:16
• ------ That's a good question. I don't know what algorithm is being used to find integral points. However, for curves of rank 1, I know that the methods used are pretty good, and can often be used to prove that the list is complete (via linear forms in elliptic logs, for example). You might try looking at the LMFDB documentation to see if they explain how reliable their lists of integral points are. Oct 6, 2017 at 13:18
• How exactly is this Weierstrass model computed (which transformation is used)? Oct 17, 2017 at 0:20
• Your equation $y^2=x^3−39x^22+320x+1024$ is not a minimal Weierstrass equation, it is not minimal at 2. If I'm reading the output from PARI correctly, a change of variables to get a minimal equation is $x\to 4x$ and $y\to 8y+4x$, although it requires another change of variables to put it in the form listed in the LMFDB. In any case, I guess the answer to the question you're really asking is that the integer points on the minimal W.E. may not give all integral pts on your curve, since you also need pts on the minimal W.E. with $x=a/4$ and $y=b/8$ with $a$ and $b$ integers. Oct 17, 2017 at 2:39

There is sage code in John Cremona's answer to the question How to find all integer points on an elliptic curve? that is worth a try in your case. Magma also has a function for this as documented in https://magma.maths.usyd.edu.au/magma/handbook/text/1455#16527. Try both and see.