Elliptic Curve - Prime Generator

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This question is in regards to the prime generator for an elliptic curve. I have taken the prime curve $\mathbb Z_p$ and have the $5$ prime number factors for the prime curve using Euler's Totient. Is there an algorithm to figure out prime generator(s) for the curve without using the naive approach for testing each integer which passes the congruency test for not being equal to $1 \mod p$?