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(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)
