# Define an elliptic curve by its equation

Suppose we take the elliptic curve $E : y^2 = (ax+1)(bx+1)(cx+1)$ where $a,b,c \in \mathbb{Z}.$ We can define it by E = EllipticCurve[a1,a2,a3,a4,a6] only if $abc = 1.$ Suppose $abc \neq 1$ then how to define it in its originality, i.e. without applying any co-ordinate transformations?

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sage: x,y,z=polygens(QQ,'x,y,z')
sage: E=EllipticCurve_from_cubic(-y*y*z+(2*x+z)*(3*x+z)*(4*x+z))
sage: E
Scheme morphism:
From: Projective Plane Curve over Rational Field defined by 24*x^3 + 26*x^2*z - y^2*z + 9*x*z^2 + z^3
To:   Elliptic Curve defined by y^2 = x^3 + 26*x^2 + 216*x + 576 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(-x : -y : -1/24*z)

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How to find integral points on this curve?

( 2022-02-03 06:52:21 +0200 )edit