How to check whether the given curve is Elliptic or not? I've given a curve E5(2,3) over Zp. How to check this curve is Elliptic or not? I know the curve formed by this will be:
(y2) mod 5 = (x3 + 2*x+3)mod 5.
For most cases, you can take this as a (simplified) definition of Elliptic Curve:
An elliptic curve over $K$ (a field) is a plane algebraic curve over $K$ defined by an equation of the form

$y^{2}=x^{3}+ax+b$    (1)

($a,b \in K$) that is non-singular (it has no cups or self-intersection).

Since the curve you give in the question is of the form $y^{2}=x^{3}+ax+b$, we just need to check the non-singularity!

This can be done by computing the discriminant

$\Delta = -16(4a^3 + 27b^2)$.

Fact: an algebraic curve defined by an equation of the form (1) is non-singular if and only if its discriminant is non-zero (this is a standard result in the theory of elliptic curves)

In our case, $\Delta = -16(4*8 + 27*9) = -16(32+243) = -16(275) = 0$ (mod 5) 

So the discriminant is zero in the field $\mathbb{Z}_5$, hence the curve is singular, hence it is not an elliptic curve.

Hope it helps, feel free to ask for further details.