Can we exploit ECDLP with isogenies between curves?

I got a situation here. I have two elliptic curves, E1 and E2, both over the same finite field. E1 is pretty secure, prime order, large embedding degree, not pairing-friendly. E2 shares those traits but totally different J-invariant and discriminant. The isogeny between them is small degree. Can we leak anything about the ECDLP through these curves?

Nah, that's not gonna work. For an isogeny between prime-order curves, the kernel has to be trivial or equal to the prime order. That would just turn the curve into a trivial group. Plus, E1 and E2 have different J-invariants, so a non-trivial isogeny can't exist. It’s basic math.

Wait, are you saying no isogeny can exist at all? That's not right. Sure, there can’t be an isomorphism, which is like a degree-1 isogeny. But other prime degree isogenies can exist even with different J-invariants.