Is there an elliptic curve mod n with exactly one point?

Question

I have tried many elliptic curves $y^2 = x^3 + ax +b$ with no success. I know that for prime modules there exists a minimum number of points the elliptic curve has to have, and I couldn't satisfy this for the smallest primes. So I decided to try luck with modules with few quadratic residues such as 8. But again, no luck.

Answer 1

The answer is: $y^2 + y = x^3 + x +1 \pmod{2}$. The only element here is the point in infinity: the neutral element.

Answer 2

You can also construct an answer mod 3 (which is in short weierstrass form) as follows:

For all $x \in \mathbf Z/3\mathbf Z$ we know by Fermat's little theorem that $x^3 = x$, therefore the polynomial $x^3 - x + 2$ always takes the value $2$ on elements of $\mathbf Z/3 \mathbf Z$, as 2 is not a square in this ring the curve

$$y^2= x^3 - x + 2$$

has no non-infinite points.