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Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ or $\mathbb{F}_q$ rational endomorphism ring, $\mathsf{End}_{{\mathbb F}_q}(E)$, i.e. those endomorphism with $\mathbb{F}_q$ coefficients. Below, I give some context on why I am interested in it.

Any reference is appreciated.

Context: The ideal class groups $\mathrm{Cl}(\mathcal{O})$ are of great interest in cryptographical scenario e.g. the Cl encryption scheme in those homomorphic encryption literatures. It also relates nearly to Couveignes-Rostovtsev-Stolbunov (CRS) based cryptography, such as CSIDH, in which we have class group randomly acts on a class of isogeneous elliptic curves. However, instead of computing the exact class group structure in the first place, to my knowledge most of them chose to indirectly compute these group actions, (e.g. using "elliptic curve tricks" such as modular polynomials, division polynomials or finding torsion points). If one could construct an elliptic curve (especially supersingular ones) together with its class group structure, then we could directly benefit from those class group technique such as (low-level) algorithmic improvements from binary quadratic form etc.

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