Function: ellztopoint
Section: elliptic_curves
C-Name: pointell
Prototype: GGp
Help: ellztopoint(E,z): inverse of ellpointtoz. Returns the coordinates of
point P on the curve E corresponding to a complex or p-adic z.
Doc:
$E$ being an \var{ell} as output by
\kbd{ellinit}, computes the coordinates $[x,y]$ on the curve $E$
corresponding to the complex or $p$-adic parameter $z$. Hence this is the
inverse function of \kbd{ellpointtoz}.
\item If $E$ is defined over a $p$-adic field and has multiplicative
reduction, then $z$ is understood as an element on the
Tate curve $\bar{Q}_p^* / q^\Z$.
\bprog
? E = ellinit([0,-1,1,0,0], O(11^5));
? [u2,u,q] = E.tate; type(u)
%2 = "t_PADIC" \\ split multiplicative reduction
? z = ellpointtoz(E, [0,0])
%3 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
? ellztopoint(E,z)
%4 = [O(11^9), O(11^9)]
? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
? z = ellpointtoz(E,[x,y]); \\ non-split: t_POLMOD with t_PADIC coefficients
? P = ellztopoint(E, z);
? P[1] \\ y coordinate is analogous, more complicated
%8 = Mod(O(2^4)*x + (2^-1 + O(2^5)), x^2 + (1 + 2^2 + 2^4 + 2^5 + O(2^7)))
@eprog
\item If $E$ is defined over the complex numbers (for instance over $\Q$),
$z$ is understood as a complex number in $\C/\Lambda_E$. If the
short Weierstrass equation is $y^2 = 4x^3 - g_2x - g_3$, then $[x,y]$
represents the Weierstrass $\wp$-function\sidx{Weierstrass $\wp$-function}
and its derivative. For a general Weierstrass equation we have
$$x = \wp(z) - b_2/12,\quad y = \wp'(z)/2 - (a_1 x + a_3)/2.$$
If $z$ is in the lattice defining $E$ over $\C$, the result is the point at
infinity $[0]$.
\bprog
? E = ellinit([0,1]); P = [2,3];
? z = ellpointtoz(E, P)
%2 = 3.5054552633136356529375476976257353387
? ellwp(E, z)
%3 = 2.0000000000000000000000000000000000000
? ellztopoint(E, z) - P
%4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
? ellztopoint(E, 0)
%5 = [0] \\ point at infinity
@eprog