The rational points of the non-split Cartan modular curve of level 121

We prove that the modular curve X⁺ₙₛ(121) of the normalizer of a non-split Cartan subgroup of level 121, of genus 511, has exactly seven rational points, all of them CM. This is the level 121 case of a conjecture of Rouse, Sutherland and Zureick-Brown, and with their theorem it determines the 11-adic images of Galois of non-CM elliptic curves over ℚ. The Jacobian has rank equal to the genus. We work at the primes 17 and 43, at each of which one Hecke block of the Mordell–Weil group has logarithm divisible by p² while its p-adic height has valuation exactly one. A comparison of valuations in the quadratic Chabauty identity then shows that the residue disc of each CM point holds no other rational point, with no computation of the local height at p, no basis of the Mordell–Weil group and no model of the curve. The inputs come from expansions of differentials at CM points and from a formula we prove for intersection numbers of CM divisors under Hecke correspondences on X⁺ₙₛ(ℓᵉ), at every prime v ≠ ℓ, and at v = ℓ when ℓ ≥ 7, ℓ ≡ 3 mod 4 and e ≤ 2.