On elliptic curves whose 3-torsion subgroup splits as \(\mu_3 \oplus \mathbb Z/ 3\mathbb Z\) (Q2904291)

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with a \(3\)-torsion point rational over \(\mathbb Q\). By translating this point to \(P=(0,0)\), one deduces that \(E\) admits a Weierstrass equation of the shape NEWLINE\[NEWLINEy^2+axy+by=x^3\quad \text{with}\quad a,b\in \mathbb Q.\tag{1}NEWLINE\]NEWLINE Its discriminant is \(b^3(a^3-27b)\). Let \(\mu_3\) be the group of the cubic roots of unity and \(E[3]\) the \(3\)-torsion subgroup of \(E\). Using the Weil pairing \(e_3 : E[3]\times E[3]\to \mu_3\), we can define a map \(E[3]\to \mu_3\) by \(Q\mapsto e_3(P,Q)\). This map is surjective and its kernel is the subgroup generated by \(P\). One obtains the exact sequence of \(\text{Gal}\biggl(\overline {\mathbb Q}/{\mathbb Q}\biggr)\)-modules NEWLINE\[NEWLINE0\to \mathbb Z/3\mathbb Z \to E[3]\to \mu_3\to 0.NEWLINE\]NEWLINE The author shows that this sequence splits, in other words, the \(\text{Gal}\biggl(\overline {\mathbb Q}/{\mathbb Q}\biggr)\)-modules \(E[3]\) and \(\mathbb Z/3\mathbb Z\times \mu_3\) are isomorphic, if and only if \(a^3-27b\in \mathbb Q^{*3}\). He then determines the Weierstrass equation of the curve \(E/\mu_3\) of the from (1). As an application, he obtains all the isogeny relations among the elliptic curves of \(3\)-power conductor which have a \(3\)-torsion point. In the final section, for a non-zero integer \(m\), the author studies the relation between the set of integral points of the curve \(x^3+y^3=m\) and the elliptic curves \(E\) over \(\mathbb Q\) such that the \(\text{Gal}\biggl(\overline {\mathbb Q}/{\mathbb Q}\biggr)\)-modules \(E[3]\) and \(\mathbb Z/3\mathbb Z\times \mu_3\) are isomorphic.