Writing a particular Elliptic Curve

[Bhavin Moriya]

How do I write the following elliptic curve in SAGE,

y^2 = A*x^3 + B*x^2 + C*x + D?

[Jeroen Demeyer]

On 2014-04-10 08:25, Bhavin Moriya wrote:
> How do I write the following elliptic curve in SAGE,
>
> y^2 = A*x^3 + B*x^2 + C*x + D?

You cannot, because Sage requires the x^3 term to have coefficient 1. If
you multiply both sides of that equation by A^2 and do a variable
substitution, you get something of the form

y^2 = x^2 + b*x^2 + c*x + d

which can be input in Sage in the usual way (see EllipticCurve?).

[Peter Bruin]

Hello,

Sage only supports equations where the coefficients of y^2 and x^3 are 1; you first need to multiply the equation by A^2 and then put A*y = Y and A*x = X, so the equation becomes

Y^2 = X^3 + B*X^2 + A*C*X + A^2*D

You can construct this (assuming A, B, C, D are set to suitable values) using

sage: E = EllipticCurve([0, B, 0, A*C, A^2*D])

I hope this answers your question.

Peter


Op donderdag 10 april 2014 07:25:15 UTC+1 schreef Bhavin Moriya:

[Bhavin Moriya]

Dear Sir,

Thank you very much for the suggestion. Could you please tell me how do I do 

a variable substitution

?

Thanks,

Bhavin.

On Thu, Apr 10, 2014 at 6:56 PM, Jeroen Demeyer <jdem...@cage.ugent.be> wrote:

do a variable substitution

Bhavin K Moriya
NASI Research Associate
Harish Chandra Research Institute

Chhatnag Road, Jhusi

Allahabad - 211 019.

"Only the gentle are ever really strong."
- James Dean

[Bhavin Moriya]

Dear Peter,

Thank you very much for your answer. It helps but still I do not know how to see the answer to my exact problem.

My exact problem is the see the graph of y^2=x(x+1)(2*x+1)/6. 

[Peter Bruin]

Dear Bhavin,

Thank you very much for your answer. It helps but still I do not know how to see the answer to my exact problem.

My exact problem is the see the graph of y^2=x(x+1)(2*x+1)/6.

If you have constructed an elliptic curve (after scaling the variables), say E, then you can do

sage: E.plot()

Otherwise, you can try something like

sage: R.<x,y>=RR[]
sage: f=y^2-x*(x+1)*(2*x+1)/6
sage: C=Curve(f)
sage: C.plot()

Peter

[Bhavin Moriya]

Thanks Peter.

Bhavin K Moriya
NASI Research Associate
Harish Chandra Research Institute

Chhatnag Road, Jhusi

Allahabad - 211 019.

"Only the gentle are ever really strong."
- James Dean

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