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Sorry for my bad English.

Let $p$ be a prime, $E$ be an elliptic curve over $\mathbb{F}_p$ of $j$-invariant is 0 or 1728.

Now I want to know if there is a criterion of when $E$ is supersingular.

In Wikipedia,there is a table for small $p$, but I don't know the regularity of connection of $p$ and supersingular.

Yos
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    The $j$-invariants $0$ and $1728$ correspond to elliptic curves with CM by $\mathbb Z[\omega]$ and $\mathbb Z[i]$ respectively so their reduction mod $p$ will be supersingular precisely at primes $p$ that are split in the respective rings. In this case $p\equiv 2 \pmod 3$ and $p \equiv 3 \pmod 4$. My favorite way to see this is that the two pieces of the newton polygon are either preserved or flipped by the Galois action in the ordinary and supersingular case respectively. – Arkady Dec 02 '22 at 08:42
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    Did you mean invariant? – J. W. Tanner Dec 02 '22 at 10:02
  • If $(p)$ is inert in $\Bbb{Z}[i]$ consider the reduction of $E:y^2=x^3+x$ modulo $p$ and $i=(-x,iy)$, then the Frobenius $\phi_p(x,y)=(x^p,y^p)$ doesn't commute with $[i] \in End(E\bmod p)$, this implies that $(E\bmod p)[p^\infty]$ can't be infinite so $E\bmod p$ is supersingular. – reuns Dec 02 '22 at 10:41
  • Conversely if $(p)$ splits in $\Bbb{Z}[i]$ then $\phi_p$ commutes with $[i]$ so $\phi_p\in \Bbb{Z}[i]$ whence it can't be that $[p]=\phi_p^2 u$ for some $u\in Aut(E\bmod p)$ and $E\bmod p$ is ordinary. – reuns Dec 02 '22 at 11:42

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