Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
2026-03-30 12:00:22.1774872022
$j$-invariants of isogenous elliptic curves
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Yes - if they are $m$-isogenous, then the modular polynomial $\Phi_{m} (j, j')$ is zero. (This is actually the definition of $\Phi_m$). If they are not $m$-isogenous, then the best I can say is that the corresponding lattices will be (rationally) commensurable, so that any representatives $\tau, \tau'$ in the Poincaré half-plane will be related by a rational homography. Since for a given $\tau$, the set of all isogenous $\tau'$ is dense in the half-plane, there cannot be any analytic function of $\tau, \tau'$ detecting the isogenies of any degree. The same goes for $j, j'$ of course.