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My question is as follows: Pick two values of a in $F_{11} = Z/11Z$ (a not equal to 3), such that the equation $y^2 = x^3+ax+1$ defines an elliptic curve (i.e., it is smooth).

For each such a, determine the number of points #E(F_11).

I'm new to elliptic curves in number theory, so any tips or solutions to this problem would be greatly appreciated!

suomynonA
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K.Garn
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  • Where are you stuck? What have you tried? Do you know how to check that a curve is non-singular? To begin, try $a=0$ and $a=1$ and see if those work. – Arkady Dec 07 '17 at 03:00
  • So I have that my n=11, and I am choosing a values at random. I am computing to see if 4a^3+27 is congruent to 1 (mod 11). If it is, then y^2=x^3+ax+1 is smooth, since 4a^3+27 is a unit in (Z/nZ). I have only found that a=6 is congruent to 1 (mod 11). Since I only have 1 value of a which is congruent to 1 mod 11, I am stuck and don't know where to go from here. Am I supposed to say that a=6 is one value, and since we are dealing in F11, a=17 is another value? – K.Garn Dec 07 '17 at 03:01
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    You just need the discriminant to be nonzero, not necessarily =1. Anything nonzero is a unit in Z/11. – Ben Dec 07 '17 at 03:29
  • I don't see what to do from here. – K.Garn Dec 07 '17 at 03:35

1 Answers1

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For $a = 0$, we have the elliptic curve

$$y^2 \equiv x^3 + 1 \pmod{11} $$

A naive way to find the number of points is to just enumerate them, we find

$$ \begin{array}{c|lcr} x & \text{y's} \\ \hline 0 & 1, 10 \\ 1 & \text{None} \\ 2 & 3, 8 \\ 3 & \text{None} \\ 4 & \text{None} \\ 5 & 4, 7 \\ 6 & \text{None} \\ 7 & 5, 6 \\ 8 & \text{None} \\ 9 & 2, 9 \\ 10 & 0 \\ \end{array} $$

We can see that we have $11$ points in that table plus the point at infinity for a total of

$$\#E(F_{11}) = 12$$

You can find a second example.

Moo
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  • I hate to do this because it's unrelated to this question, but this is the only way I can see to contact you. You provided a crucial comment to a question that has since been marked as deleted. I posted a follow up question to get more info. Do you think you could take a look at this question? Thank you!! https://math.stackexchange.com/questions/2866222/what-are-the-steps-involved-in-solving-a-modular-quartic-polynomial-equation – Levitikon Jul 30 '18 at 11:07