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The potential of the $BC_1$ quantum elliptic model is a superposition of twoWeierstrass functions with doubling of both periods (two coupling constants).The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized bythe Lam\'e Hamiltonian. It is shown that in the space of $BC_1$ ellipticinvariant, the potential becomes a rational function, while the flat spacemetric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra forarbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in threedifferent magnetic fields. It is shown that there exist three one-parametricfamilies of coupling constants for which a finite number of polynomialeigenfunctions (up to a factor) occur.

The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions