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October 26th, 2014, 12:26 AM   #1
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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?
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October 27th, 2014, 06:59 PM   #2
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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.
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October 27th, 2014, 08:13 PM   #3
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Quote:
Originally Posted by Omnipotent View Post
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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