Defining an elliptic curve by a cubic equation

Question

Suppose we have an elliptic curve $y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6$, we define it on SAGEMATH by E = EllipticCurve([a1,a2,a3,a4,a6]). Suppose the elliptic curve is given in a cubic equation form where the leading co-efficient is not equal to $1$, e.g., $~y^2 = (2x+1)(3x+1)(4x+1)$. Then how can we define this curve in SAGEMATH in its original form (not taking any isomorphism). Also how can we find all the integer points on it?

Answer 1

Homogenizing its original curve, we get $ Y^2 Z = \left( 2X +Z \right) \left( 3X+Z \right) \left( 4X+Z \right) $. Performing the change of coordinates $ Z = 24 Z' $ e dehomogenizing the coordinate $ Z' $, we get $$ E' : y'^2 = \left( x' + 12 \right) \left( x' + 8 \right) \left( x' + 6 \right), $$ where $ \left( x', y' \right) = \left( \frac{X}{Z'}, \frac{Y}{Z'} \right) $.

Then $$ \left( x,y \right) = \left( \frac{X}{Z}, \frac{Y}{Z} \right) = \left( \frac{X}{24Z'}, \frac{Y}{24Z'} \right) = \left( \frac{x'}{24}, \frac{y'}{24} \right).$$

Therefore, we are looking for points integers in $ E' $ such that, both the coordinates are divisible by 24.

Running the SageMath code:

R.<x,y> = QQ[]
E = EllipticCurve(y^2 - (x + 12)*(x + 8)*(x + 6))
E.integral_points(both_signs=True)

it return all integers points in $ E'$ (with coordinates $ \left[ X:Y:Z' \right] $):

[(-12 : 0 : 1), (-10 : -4 : 1), (-10 : 4 : 1), (-9 : -3 : 1), (-9 : 3 : 1), (-8 : 0 : 1), (-6 : 0 : 1), (-4 : -8 : 1), (-4 : 8 : 1), (0 : -24 : 1), (0 : 24 : 1), (42 : -360 : 1), (42 : 360 : 1)]

Therefore, the only the integers points with both the coordinates divisible by 24 are $ \left( x', y'\right) = \left( 0 , \pm 24 \right) $.

Finally, the only integer points in our original curve are $ \left( 0, \pm 1 \right) $.