I am working something where a picture like this one appeared :
Say the curve is written in the form $ y^2 = x^3 + ax^2 + bx + c $ (if this is the wrong form of coefficients, feel free to correct me, I am guessing here) what are the conditions on $a,b,c$ so that the elliptic curve "looks like the one in red in the picture"? Essentially I'm asking this because I had an elliptic course that I can't recall (because my teacher barely gave course notes at all... so I have no reference), but I remember the pictures and I was doing something in graph theory where this precise picture showed up, so I want to try to fit an elliptic curve to the curve I have and see if my conjecture that the curve is elliptic is morally valid.
Thanks in advance!