Hyperelliptic $$\mathcal {S}_7$$-curves of prime conductor

Abstract

An abelian threefold $$A_{/{\mathbb {Q}}}$$ of prime conductor N is favorable if its 2-division field F is an $${\mathcal {S}}_7$$ -extension over $${\mathbb {Q}}$$ with ramification index 7 over $${\mathbb {Q}}_2$$ . Let A be favorable and let B be a semistable abelian variety of conductor $$N^d$$ with B[2] filtered by d copies of A[2]. We obtain a class field theoretic criterion on F to guarantee that B is isogenous to $$A^d$$ and a fortiori, A is unique up to isogeny.