Local solubility of the cubic x cubed minus x minus one ($y^2=x^3-x-1\text{ over }{\mathbb{Z}}_p⟺p\ne 3$)

Modulo three the right side is always minus one, a nonsquare. At two, $x=-2$ gives the square 2-adic integer $-7\equiv 1(\mathrm{mod}8)$. At twenty-three the same value is a nonzero square modulo the prime and lifts by Hensel's lemma. At every other prime at least five, the model has good reduction of an elliptic curve and the Hasse bound gives more than its single point at infinity, hence an affine smooth point. Lifting that point gives p-adic integer coordinates.