$${{\mathbb {Q}}}$$-curves over odd degree number fields

By reformulating and extending results of Elkies, we prove some on \({{\mathbb {Q}}}\)-curves over number fields odd degree. We show that, such fields, the only prime isogeny degrees \(\ell \) that a {Q}}}\)-curve without CM may have are those already possible {Q}}}\) itself (in particular, \le 37\)), existence bound cyclic isogenies between depending degree field. also torsion groups not divisible by 7\) 15 appear as elliptic curves {Q}}}\). Complementing these theoretical results, give an algorithm for establishing whether any given curve E is {Q}}}\)-curve, involves working {Q}}}(j(E))\).