Re: question on mapping points from an elliptic curve back to a quartic

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Oct 09, 2024 at 07:31:35PM +0200, Bill Allombert wrote:
> On Wed, Oct 09, 2024 at 09:54:10AM -0700, American Citizen wrote:
> > Suppose we consider a quartic with rational points
> > 
> >   Q(x,y) : -x^4 + 39/380*x^3 + 39/380*x + 1 = y^2
> > 
> > Question:
> > 
> >   Why are most of the points in the elliptic curve pool L unmappable back to
> > Q(x,y)? This is surprising to me, as I believed that all the rational points
> > on E were mappable back to Q(x,y)?
> 
> Q is a 2-cover of E, so only the points in [2]E(\Q) are mappable back to Q.

And to answer your question quantitatively:

E ~ Z^3 x Z/2Z  
[2]E ~ (2Z)^3 x 0Z/2Z

E/[2]E ~ (Z/2Z)^4

[E:[2]E]=16 so a point on E has proba 1/16 to be mappable back to Q.

Cheers,
Bill