Furio–Lombardo Conjecture 1.6

Furio and Lombardo’s 2025 work reduces the remaining case to determining the rational points on a specific plane quartic. The conjecture asserts that this quartic has exactly four rational points, which would complete the relevant $7$-adic classification.

Known results

• Furio and Lombardo, July 23, 2025: proved that $X_{ns}^{+}(49)$ has exactly seven rational points, all corresponding to CM elliptic curves.
• Furio and Lombardo, July 23, 2025: reduced the complete classification of $7$-adic images to Conjecture 1.6, namely the rational-point computation on the remaining plane quartic.

September 28, 2026 claimed solution

A preprint by Tho Nguyen Xuan states that it solves Furio–Lombardo Conjecture 1.6 and completes the $7$-adic classification for non-CM elliptic curves over $\mathbb{Q}$. This is an unreviewed solitary claim, with no retrieved independent verification.

Current status (as of September 2026): the earlier partial results are established, while the claimed solution of Conjecture 1.6 and the resulting complete classification remain unverified.