In recent years, significant progress has been made on Mazur's Program B, with many authors beginning a systematic classification of all possible images of p-adic Galois representations attached to elliptic curves over Q. Currently, the classification is only complete for p \2, 3, 13, 17\. The main difficulty for other primes arises from the need to understand elliptic curves whose mod-pⁿ Galois representations are contained in the normaliser of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves X₍ₒ^+ (pⁿ). Here, we consider the case p=7 and show that the modular curve X₍ₒ^+ (49), of genus 69, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on X₍ₒ^+ (49) and the primitive integer solutions of the generalised Fermat equation a² + 28b³ = 27 c⁷, the resolution of which can be reduced to determining the rational points of several genus-three curves. Furthermore, we reduce the complete classification of 7-adic images to the determination of the rational points of a single plane quartic.
Furio et al. (Wed,) studied this question.