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Prime ideals above two and three in the cubic field of discriminant minus 307

For

g (X) = X^{3} - 5 X^{2} + 11 X - 12

, the reductions are

\overline{g} = X (X^{2} + X - 1) over F_{3}

and

\overline{g} = X (X^{2} + X + 1) over F_{2} .

Both quadratic factors are irreducible, so

(3) = (3, β) (3, β^{2} + β - 1)

and

(2) = (2, β) (2, β^{2} + β + 1) .

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