Prime ideals above two and three in the cubic field of discriminant minus 307
= Prime ideals above two and three in the cubic field of discriminant minus 307
For $g(X)=X^3-5X^2+11X-12$, the reductions are
$$
\overline g=X(X^2+X-1)\quad\text{over }\mathbb F_3
$$
and
$$
\overline g=X(X^2+X+1)\quad\text{over }\mathbb F_2.
$$
Both quadratic factors are irreducible, so
$$
(3)=(3,\beta)(3,\beta^2+\beta-1)
$$
and
$$
(2)=(2,\beta)(2,\beta^2+\beta+1).
$$