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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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2 Answers 2

up vote 2 down vote accepted

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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Thanks! Very clear. +1 & check! –  Patrick Da Silva Sep 9 '12 at 1:06
    
It seems to be that this is a sufficient but not necessary condition. It only describes curves where the crossing is at $(0,0)$, not their translations to the sides. –  Henning Makholm Sep 9 '12 at 12:00
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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. –  Patrick Da Silva Sep 10 '12 at 7:53

All you need is a repeated zero. $y^2=x(x-1)^2$ will do.

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Can that give me possibly a curve without a loop? Or do I need a triple zero for that? I guess I should've thought of it, it's quite clear that the curve passes twice through the point... I guess I just answered myself ; I'll have the loop if I have one zero away from the double zero. Thank you! –  Patrick Da Silva Sep 9 '12 at 0:26
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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 =) –  Patrick Da Silva Sep 9 '12 at 1:05

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