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Shouldn't Schoof's algorithm be taking generator point as an input ? All implementations I tried (including Sage) need only the coefficients of the Weierstrass equation and a prime. No mention of the generator point at all.

For elliptic curve y2 = x3 + 7 MOD 199, generator (11,12) gives a total of 63 points (including the point at infinity). While for same elliptic curve, generator (5, 27) gives a total of 21 points.

user1035924
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    Schoof's algorithm doesn't need the generator as an input because the algorithm wants to calculate the total number of points rather only the number of points in the cyclic subgroup generated by some point. The elliptic curve $y^2=x^3+7$ with $x,y\in\Bbb{F}_{199}$ has a total of $189$ points (including the point at infinity), so Schoof should answer $189$. Observe that both $21$ and $63$ are factors of $189$ as they should by Lagrange's theorem. – Jyrki Lahtonen Sep 17 '23 at 19:59
  • Now I understand. Thank you so much. – user1035924 Sep 20 '23 at 12:17

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