Function: ellheegner
Section: elliptic_curves
C-Name: ellheegner
Prototype: G
Help: ellheegner(E): return a rational nontorsion point on the elliptic curve E
 assumed to be of rank 1.
Doc: Let $E$ be an elliptic curve over the rationals, assumed to be of
 (analytic) rank $1$. This returns a nontorsion rational point on the curve,
 whose canonical height is equal to the product of the elliptic regulator by the
 analytic Sha.

 This uses the Heegner point method, described in Cohen GTM 239; the complexity
 is proportional to the product of the square root of the conductor and the
 height of the point (thus, it is preferable to apply it to strong Weil curves).
 \bprog
 ? E = ellinit([-157^2,0]);
 ? u = ellheegner(E); print(u[1], "\n", u[2])
 69648970982596494254458225/166136231668185267540804
 538962435089604615078004307258785218335/67716816556077455999228495435742408
 ? ellheegner(ellinit([0,1]))         \\ E has rank 0 !
  ***   at top-level: ellheegner(E=ellinit
  ***                 ^--------------------
  *** ellheegner: The curve has even analytic rank.
 @eprog
