plot of adding points on an elliptic curve

ec.svg

ec.tex

\documentclass{minimal}  
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[yscale=0.5]
			%TODO: up sample size
			\draw[domain=-4.060646:2,samples=100,smooth,variable=\x,blue] plot ({\x},{sqrt(\x^3 + 4*(\x)^2 + 1)});
			\draw[domain=-4.060646:2,samples=100,smooth,variable=\x,blue] plot ({\x},{-sqrt(\x^3 + 4*(\x)^2 + 1)});
			\node (P) at (-3.5, -2.66926956300783) {};
			\node (Q) at (-0.25, 1.1102430216445) {};
			\node (R) at (1.10295826812394, 2.68474117626385) {};
			\node (P+Q) at (1.10295826812394, -2.68474117626385) {};
			\draw[color=red,shorten >=-1cm,shorten <=-1cm] (P) -- (R);
			\draw[color=red,shorten >=-1cm,shorten <=-1cm] (R) -- (P+Q);
			\node[circle,draw=black,fill=green,label=above:{$P$}] at (P) {};
			\node[circle,draw=black,fill=green,label=above:{$Q$}] at (Q) {};
			\node[circle,draw=black,fill=green] at (R) {};
			\node[circle,draw=black,fill=green,label=left:{$P+Q$}] at (P+Q) {};
\end{tikzpicture}
\end{document}

plot.sage

import matplotlib.pyplot as plt
plt.rc('text', usetex=True)
plt.rc('font',**{'family':'serif','serif':['Computer Modern Roman'],'size':14})
E = EllipticCurve([0,4,0,0,1])
P = E.change_ring(RR).lift_x(-0.25)[:2]
Q = E.change_ring(RR).lift_x(-3.5,all=true)[1][:2]
R2 = (E.change_ring(RR)(P) + E.change_ring(RR)(Q))[:2]
R1 = [i*x for i,x in zip(R2,[1,-1])]
L1 = [s*(vector(P) - vector(Q)) + vector(Q) for s in [-100,100]]
L2 = [s*(vector(R1) - vector(R2)) + vector(R2) for s in [-100,100]]

p = E.plot(axes=false,zorder=0)
p += line(L1,color="red",zorder=1)
p += line(L2,color="red",zorder=1)
p += point(P,size=100,zorder=2,color="green")
p += text("$\mathcal{P}$",[P[0]-0.5,P[1]],horizontal_alignment="left",color="black",fontsize=20)
p += point(Q,size=100,zorder=2,color="green")
p += text("$\mathcal{Q}$",[Q[0]-0.5,Q[1]],horizontal_alignment="left",color="black",fontsize=20)
p += point(R1,size=100,zorder=2,color="green")
p += text("$\mathcal{P} * \mathcal{Q}$",[R1[0]-0.8,R1[1]] ,horizontal_alignment="left",color="black",fontsize=20)
p += point(R2,size=100,zorder=2,color="green")
p += text("$\mathcal{P} + \mathcal{Q}$",[R2[0]+0.25,R2[1]],horizontal_alignment="left",color="black",fontsize=20)
p.show(xmin=-4.25,xmax=2,ymin=-4,ymax=4)