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I am working something where a picture like this one appeared : Elliptic Curve

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!

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If applying to your original equation for the curve

$ y^2=x^3+ax^2+bx+c $

then the curves similar to the red curve appear when $b$ and $c$ are $0$. The coefficient on $x^2$ can be anything within $(0,\infty)$. For an equation where the curve can be translated, consider

$ (y-a)^2=(x-b)(x-c)^2 $

with conditions that $b-c\gt 0$. The variable $a$ is the vertical translation of the curve, $b$ is the $x$-coordinate of the intersection, and $c$ is the $x$-coordinate of the loop. This is also why Gerry's answer worked, as $ y^2=x(x-1)^2 $ is of this form as well.

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    @Envious : Gerry's answer pretty much gave me the hint I needed... I did an elliptic curve course, so he just made me remember with the quick example that since the curve goes "twice" through the point there has to be a double root. But I'm still glad you worked it out.2012-09-10
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All you need is a repeated zero. $y^2=x(x-1)^2$ will do.

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    Your answer helped me very quickly but I guess Envious Page needs the reuptation more than you do, and his answer is still as good as yours. I'll give him the check but you both get +1 =)2012-09-09