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Please help or hints me to solve this question:

Suppose that $E_1: y^2= x^3+2x+5$ and $E_2: y^2= x^3+3x+5$ are two elliptic curve on $\Bbb F_{361}$. Show that #$E_1(\Bbb F_{361})$=#$E_2(\Bbb F_{361})$.

user26857
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Masoud
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1 Answers1

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$(7x)^3+3(7x)+5 \equiv x^3+2x+5 \pmod {19}$

mercio
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  • iirc, it's enough to compare their $j$ invariants mod $19$, because unless $j=0$ or $j=1728$, any isogeny between the two is defined at worst over the quadratic extension of $\Bbb F_{19}$, and this forces them to have the same number of points in $\Bbb F_{19^2}$. But heh, since here the isogeny is defined over $\Bbb F_{19}$ anyway... – mercio May 27 '17 at 18:46