The rational points of the non-split Cartan modular curve of level 25

We prove that the modular curve X⁺ₙₛ(25) of the normalizer of a non-split Cartan subgroup of level 25, of genus 14 and Jacobian rank 14, has exactly nine rational points, all of them CM. This is the level 25 case of a conjecture of Rouse, Sutherland and Zureick-Brown, and with their theorem it determines the 5-adic images of Galois of elliptic curves over ℚ without complex multiplication. A descent by the modular units of the curve, using no information on the Mordell–Weil group, places every rational point in the residue disc of a CM point at the prime 1151. Quadratic Chabauty at 1151, with two correspondences that change only the level structure, then shows that each of these discs holds one rational point. The local term at 5 of these correspondences vanishes, by a formula of Betts and Dogra, because their eigenvalues lie in √5 ℚ, and the height pairing is fixed by the nine CM points alone.