Something is very wrong, perhaps just the typing of the polynomial. With gp-pari, I do get irreducible $\pmod p$ for primes $p \equiv \pm 3 \pmod {10}.$ 

Pari says the discriminant is $2^{12} \cdot 3^2 \cdot 5^3.$

It gives four linear factors for 
$$ p \in \{  29, 31, 59, 61, 89, 149, 151, 179, 181, 211, 239, 241, \ldots \}  $$

It gives two quadratic factors for 
$$ p \in \{  11,19,41,71,79,101,109,131,139,191,199,229,251, \ldots, 409,\ldots  \}  $$