How to model a miniature elliptic curve? For educational purposes I would need to work on an elliptic curve that has a small field, but holds the safety futures of a real curve. Is that possible to have such a curve!? 

For example for the curve of secp256k1, is that possible to find small $P$, $N$ and $G$ so that the mini curve does still have safety features of the real curve? By safety features I mean those features that come from the difficulty of solving the discrete logarithm problem. Of course I know that a curve with a small field is vulnerable through brute-force attacks.

To do so, first I read about properties of the prime $P$. It seems that it is a SAFE prime which is equal to $3 \bmod 4$. If I search for a small prime with the same two properties am I getting the correct mini curve?

How are $G$ and $N$ decided then?

I need a curve that lets me do point manipulations (adding, doubling, halving,...)  with plotable easy-to-work-with integers, not those 78-digits of secp256k1.
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