= Isogeny Selmer group
{title2=$\operatorname{Sel}^{\phi}(E/K)$}
For an <isogeny of elliptic curves> $\phi:E\to E'$ over a <number field> $K$, this finite subgroup of $H^1(K,E[\phi])$ consists of <Galois cohomology> classes whose restriction at every place comes from $E'(K_v)/\phi E(K_v)$. It fits into
$$
0\to E'(K)/\phi E(K)\to\operatorname{Sel}^{\phi}(E/K)\to\operatorname{Sha}(E/K)[\phi]\to0,
$$
where the last group is the kernel of the induced map on <Tate–Shafarevich groups>. Thus local computations bound the rational-point quotient without necessarily determining it. For multiplication by $n$, this recovers the <Selmer group of an elliptic curve>.
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