| American Citizen on Thu, 10 Oct 2024 20:50:11 +0200 |
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
| Re: question on mapping points from an elliptic curve back to a quartic |
Okay Bill and John:
BIll, thank you for your short explain, this is the first time that I actually understood what is going on.
John, I did read your book, but did NOT understand this part until now.
and to all, I actually started with the quartic, not the elliptic
curve, there were two GP-Pari commands available which found E,
quartic_to_ellmap() and ellfromeqn(). I used the first command to
find E, since I also needed the forward and inverse maps for the
points.
I did NOT know that Q was a two-cover for E.
Does this mean that any E found for a given Q forces (better word choice??) the Q to be a 2-cover when any GP-Pari command discovering E is used on the quartic?
Can I safely assume that I have to multiply the points on E by 2, before mapping back to Q? That's what I guess from what has been shared here from Bill
And thank you, both, for you reply.
Randall
Following on from Bill's reply, the image of the rational points on the quartic curve (which is a 2-covering of E) is one coset of 2E(Q) in E(Q). That is if the quartic has any rational points at all. That is why two-descent works: the number of 2-covers (up to appropriate equivalence) with rational points is equal to the number of such costs, i.e. the order of E/2E. That is exactly how mwrank works.
Randall, I thought you had read my book which explains this!
John
On Wed, 9 Oct 2024, 18:49 Bill Allombert, <Bill.Allombert@math.u-bordeaux.fr> wrote:
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