Function: ellheegner
Section: elliptic_curves
C-Name: ellheegner
Prototype: G
Help: ellheegner(E): return a rational non-torsion 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 non-torsion 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