= Three-isogeny descent
Let $\phi:E\to E'$ be a degree-three <isogeny> over a <field> $K$ of <characteristic> different from three. Its connecting map injects
$$
E'(K)/\phi E(K)\longrightarrow H^1(K,E[\phi]).
$$
To identify the target with $K^\times/(K^\times)^3$, one needs an identification of the <Galois module> $E[\phi]$ with $\mu_3$. This occurs, for example, when the <kernel of an isogeny> of the <dual isogeny> has a rational generator: the perfect isogeny pairing induced by the <Weil pairing> then identifies $E[\phi]$ with the homomorphisms from that constant cyclic <group> to $\mu_3$. <Kummer theory> gives $H^1(K,\mu_3)=K^\times/(K^\times)^3$; this cohomological statement does not require $\mu_3\subset K$.
Over a <number field>, the resulting classes in the <cube-class group of a field> are unramified outside finitely many places, including primes over three and primes of <bad reduction>. Their <valuations> away from that finite <set> are divisible by three, placing the image in an <S-unramified power class group>. Local image conditions then define the corresponding <isogeny Selmer group>. Without the kernel identification, one must retain the possibly twisted <Galois module> in the cohomology target.
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