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 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.

Using quartic_to_ellmap:

? [E,f,g]=quartic_to_ellmap(-x^4 + 39/380*x^3 + 39/380*x + 1);
? R=ellrank(E,10)[4]
%2 = [[6849/361,15636267/137180],[68490000/17497489,50745080927115/1390647933253],[91918208204904964/97434336836799169,101298745897513561302124420799/5778586934871496820571475070]]
? g(ellmul(E,R[1],2))
%3 = [[0,-1],[39/760,579121/577600]]~
? g(ellmul(E,R[2],2))
%4 = [[-256452291/422149780,1572345729763867/1782104367540484],[364980/536731,-271505562700/288080166361]]~
? g(ellmul(E,R[3],2))
%5 = [[-11980649304552234934739/14010204923144164009700,54935192891864148751853563818704902077136269/98142920994246485282928751383946390847045000],[12682905996665336329540/13377806069266790387539,-109870385783728297503707127637409804154272538/178965695226911372892211369459265209806476521]]~
? g(ellmul(E,R[1],1))
%6 = []~
? g(ellmul(E,R[2],1))
%7 = []~
? g(ellmul(E,R[3],1))
%8 = []~

Cheers,
Bill