ellisogeny

Function: ellisogeny
Section: elliptic_curves
C-Name: ellisogeny
Prototype: GGD0,L,DnDn
Help: ellisogeny(E, G, {only_image = 0}, {x = 'x}, {y = 'y}): compute the image
 and isogeny corresponding to the quotient of E by the subgroup G.
Doc:
 Given an elliptic curve $E$, a finite subgroup $G$ of $E$ is given either
 as a generating point $P$ (for a cyclic $G$) or as a polynomial whose roots
 vanish on the $x$-coordinates of the nonzero elements of $G$ (general case
 and more efficient if available). This function returns the
 $[a_1,a_2,a_3,a_4,a_6]$ invariants of the quotient elliptic curve $E/G$ and
 (if \var{only\_image} is zero (the default)) a vector of rational
 functions $[f, g, h]$ such that the isogeny $E \to E/G$ is given by $(x,y)
 \mapsto (f(x)/h(x)^2, g(x,y)/h(x)^3)$.
 \bprog
 ? E = ellinit([0,1]);
 ? elltors(E)
 %2 = [6, [6], [[2, 3]]]
 ? ellisogeny(E, [2,3], 1)  \\ Weierstrass model for E/<P>
 %3 = [0, 0, 0, -135, -594]
 ? ellisogeny(E,[-1,0])
 %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]
 @eprog