Let $E$ be an elliptic curve with affine equation $$y^2 = x^3+ax+b$$ where $a,b \in \Bbb Z$. Let $P$ be the set of prime numbers not dividing $ab$.
Can we prove directly that if $p \in P$ is at least $5$, then $E(\Bbb F_p) \neq \{ [0:1:0] \}$ ? This should follow from Hasse's bound $$|E(\Bbb F_p)| \geq p+1-2\sqrt p > 1, \quad\forall p \geq 5$$ But my question is about a much weaker statement, so there is probably an easier proof.