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 October 26th, 2014, 12:26 AM #1 Member   Joined: Sep 2014 From: Cambridge Posts: 64 Thanks: 4 elliptic curve symmetry and derivative Hi, I was given the formula: y^2 = x^3 +ax +b ; where a and b are constants. For the first question, it asks me to prove that the elliptic curve is symmetric about the x-axis. I drew a diagram and my answer is that all points on the curve are the same total distance away from both foci. Therefore, if (x,y) is flipped about the x axis, it becomes (x, -y) and it is also on the curve, and the same distance away from both foci. For the second question, it asks me to find dy/dx for the elliptic curve formula provided. I come to dy/dx = (3x+a) / (2y) Are my answers correct? If not how can I approach this differently?
 October 27th, 2014, 06:59 PM #2 Global Moderator   Joined: May 2007 Posts: 6,496 Thanks: 579 First question: for each x there are two values of which are equal in magnitude and opposite sign. Second question: the answer is incomplete. You should substitute the formula for y into the expresion, noting that there are two values, so the derivative has two values - not surprising since the curve has two branches. Thanks from topsquark
October 27th, 2014, 08:13 PM   #3
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Quote:
 Originally Posted by Omnipotent For the second question, it asks me to find dy/dx for the elliptic curve formula provided. I come to dy/dx = (3x+a) / (2y)
Typo? $\displaystyle \frac{dy}{dx} = \frac{3x^2 + a}{2y}$
(Note the "2" on the power of x.)

-Dan

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