I have a [hypothetical] prime $\phi$ which satisfies the divisibility conditions $$\phi \mid (2a^2-1)$$ $$\phi \mid (b^2-2)$$ $$\phi \mid (c^2-8)$$ $$\phi \mid (2d^2-289)$$ $$\phi \mid (e^2-1682)$$ $$\phi \mid (f^2-29),$$ where $a,b,c,d,e,f$ are positive integers.
What is the optimal way to determine the set of primes that $\phi$ could be drawn from? I’m assuming quadratic reciprocity is involved…?