American Citizen on Tue, 05 Dec 2023 16:43:14 +0100


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getting strange ellheight() error for number field point and curve


Hi:

Why is this happening?

I input a curve, find the minimal model of it, then carefully find an algebraic point on it.

I then find all the isogenous curves and work to move this point to all of them, and carefully check that the algebraic points are on those curves too.

All this is fine and well.

But then when I attempt to find the height of a point, things blow up.

=== gp pari script ===

minimal m[1..5] = [1, 0, 0, -114223080, -283150929600]
pt=[29099, 4571616.001890359167852374954252793155119185143350555006071823347634514965562343989206397255762390603970823616017868766947483185076860275462218345407187946353537171824424767170799129381839037]
y=[-29099/2, 1/2, 84131656042917]
which is y = -29099/2 + 1/2 * sqrt(84131656042917)
p=[29099, Mod(1/2*a - 29099/2, a^2 - 84131656042917)]
point is on curve m [1, 0, 0, -114223080, -283150929600] --> 1
F[1] = [0, 0, 0, -5482707841/48, -244634179112639/864]
P[1] = [349189/12, Mod(1/2*a, a^2 - 84131656042917)]
is on curve: 1

this works out good for all 8 curves and the point on the curve.

NF = nfinit(a^2 - 84131656042917)
FN=ellinit(F[1],NF);
Then we try to find the height as ellheight(FN,pt)
  ***   at top-level: find_iso_hts(e)
  ***                 ^---------------
  ***   in function find_iso_hts: ... ellheight(FN,pt)");H[i]=
  ***   ellheight(FN,P[i]);pr
  ***   ^---------------------
  *** ellheight: impossible inverse in Fl_inv: Mod(0, 2).
  ***   Break loop: type 'break' to go back to GP prompt
break>

so why can't we find the height of a good point on the number field initialized elliptic curve using this code?

Randall