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Define an elliptic curve by its equation

asked 2022-02-02 16:21:52 +0200

Artatrana gravatar image

updated 2022-04-14 10:52:19 +0200

FrédéricC gravatar image

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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answered 2022-02-02 16:50:40 +0200

FrédéricC gravatar image

Like this

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?

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

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Asked: 2022-02-02 16:21:52 +0200

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Last updated: Feb 02 '22