Finding integral points on an elliptic $y^2-3y=x^3+x^2$ curve using the LMFDB-database

Question

I have the following elliptic curve that I want to look up in the LMFDB-database:

$$\text{k}:\space\space\space y^2-3y=x^3+x^2$$

Using the Weierstrass form of my elliptic curve, I wrote my equation in the form:

$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$

Which gives $a_1=0,a_2=1,a_3=-3,a_4=0$ and $a_6=0$. Then I will get two parameters of this elliptic curve:

• Discriminant: $$\Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6$$

Where $b_2=a_1^2+4a_2,b_4=2a_4+a_1a_3,b_6=a_3^2+4a_6$ and $b_8=a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2$.

Using my values I get:

$$\Delta=-2331$$

• j-invariant:

$$\text{j}=\frac{\left(b_2^2-24b_4\right)^3}{\Delta}$$

Using my values I get:

$$\text{j}=-\frac{4096}{2331}$$

But when I tried to look up my elliptic curve using the discriminant I didn't find mine.

Where did I go wrong?

Answer 1

I put the sequence of Weierstrass coefficients $[a_1,a_2,a_3,a_4,a_6]=[0,1,-3,0,0]$ into the LMFDB search engine, and it gave the link to the curve $$ y^2+y=x^3+x^2+2.\qquad (*) $$ Apparently this is the minimal Weierstrass model. If you substitute $y\mapsto y-2$ into $(*)$ you get your equation back. Anyway inverting that subsitution gives you the list of integer points on your curve $$ (-1,3), (0,3), (9,30) $$ and the negatives of those points.

My very limited experience with using LMFDB suggests that typing in the Weierstrass coefficients is the simplest way. It will automatically show the minimal Weierstrass model isomorphic to the curve I input.