I would like to find an ideal $I=(f,g)\subset k[x_0,x_1,x_2]$ with $V(I)\subset \mathbb{P}^2$ having 6 points where $\deg f=2$ (which is the homogeneous defining polynomial of an ellipse) and $\deg g=3$ (which is a homogeneous curve intersecting f in 6 points) and $f,g$ is a regular sequence.
I am a beginner in Algebraic Geometry and I have the shape of $V(f)\cup V(g)$ but I do not know how to find the ideal $I$
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Later note: I need the ideal to be the intersection of the defining ideals of the six intersecting points. I mean if $_1,…,_6∈ℙ_2$ are the intersecting points of the above two curves then $=∩_{i=1}^6_{_}$ , where $_{_}=(__−__:≠0,=1,2,3)$ . Can one find such complete intersection ideal? Is there any general formula for that?