How to model a miniature elliptic curve?
The curve parameters include:

$p$: The prime modulus of the underlying field

$a$ and $b$: These are used in the curve equation, $y^2 = x^3 + ax + b$

$G$: a point on the curve that generates a subgroup

$n$: the order (number of points) of that subgroup

Once you have decided on your $p$, the process is basically try different curve parameters and a point-counting algorithm to find a curve with a large, prime order subgroup.

You can do this all automatically using SageMath

Here is a script that will generate a curve for you with the given bit size: https://github.com/brichard19/ecdl/blob/master/scripts/gencurve.sage