# $Source: /u/maple/research/lib/algcurves/src/RCS/j_invariant,v $
# $Notify: hoeij@sci.kun.nl $

macro(
	j_invariant=`algcurves/j_invariant`
):

macro(
	g_normal=`algcurves/g_normal`
):

# Compute the j invariant for curves with genus 1.
j_invariant:=proc(F,x,y)
	local f,i,c;
	global `algcurves/x0`,`algcurves/y0`;
	# global, because then the options remember in the other
	# procedures work

	options `Copyright (c) 1996 Waterloo Maple Inc. All rights reserved.`;

	readlib(`algcurves/g_expand`): # to read g_normal
	f:=collect(F,{x,y},'distributed',g_normal);
	if degree(f,y)<2 then
		ERROR(`genus is not 1`)
	elif degree(f,y)=2 then
		# We could call `algcurves/Weierstrassform` but the following
		# is likely to be faster:
		f:=collect(f/lcoeff(f,y),y,g_normal);
		f:=collect(subs(y=y-coeff(f,y,1)/2,f),y,g_normal);
		f:=coeff(f,y,0);
		# Take the factors with odd multiplicity
		f:=numer(f)*denom(f);
		f:=collect(convert([seq(`if`(irem(i[2],2)=0,1,i[1]),
		 i=evala(Sqrfree(f),'expanded')[2])],`*`),x,g_normal);
		if not member(degree(f,x),{3,4}) then
			ERROR(`genus is not 1`)
		fi;
		for i from 0 to 4 do c[i]:=coeff(f,x,i) od;
		g_normal(

256*(c[2]^2+12*c[4]*c[0]-3*c[3]*c[1])^3/(-4*c[2]^3*c[0]*c[3]^2-4*c[2]^3*c[4]*c[
1]^2+16*c[2]^4*c[4]*c[0]-80*c[3]*c[2]^2*c[1]*c[4]*c[0]+144*c[4]^2*c[1]^2*c[0]*c
[2]-192*c[4]^2*c[0]^2*c[3]*c[1]+18*c[0]*c[3]^3*c[2]*c[1]-6*c[0]*c[3]^2*c[4]*c[1
]^2+144*c[0]^2*c[3]^2*c[4]*c[2]+c[3]^2*c[2]^2*c[1]^2-128*c[4]^2*c[0]^2*c[2]^2-
27*c[0]^2*c[3]^4-27*c[4]^2*c[1]^4+256*c[4]^3*c[0]^3-4*c[3]^3*c[1]^3+18*c[3]*c[2
]*c[1]^3*c[4])

		)
	elif degree(f,x)=2 then
		procname(f,y,x)
	else
		readlib(`algcurves/Weierstrassform`)(f,x,y,
		 `algcurves/x0`,`algcurves/y0`,`j invariant`)
	fi
end:

#savelib ('j_invariant','`algcurves/j_invariant.m`'):