So you want to factor a ~1024 bits semiprime number. I tell you, RSA-330 is your friend. Here's why. Back in 1971, Daniel Shanks made a fascinating observation: sometimes factoring 2N can actually be easier than factoring N (see quoted tweet). His algorithm uses binary quadratic forms and imaginary quadratic class groups, and its difficulty depends partly on the class number of the associated quadratic field. But here's the really interesting part: you can estimate that class number without knowing the prime factors of N, using quadratic characters and a truncated Euler product. This effectively gives you a numerical index that predicts how expensive Shanks's factoring algorithm will be. Two semiprimes with the same number of bits can have significantly different predicted computational costs. So I got curious and computed this index for all 54 RSA Factoring Challenge numbers, including those factored decades ago. One number stood out: RSA-330, a 1094 bit semiprime. For the discriminant D = −N, its estimated quadratic L value is just 0.26417, giving a normalized Shanks index of approximately 0.1321, the lowest among all 54 numbers I tested. That's around the 0.3rd percentile of a simulated random quadratic character distribution for its congruence class. In other words, RSA-330 appears unusually favorable for Shanks's algorithm compared with numbers of similar size. It gets even more interesting. Since RSA-330 is a semiprime satisfying N ≡ 3 (mod 8), genus theory tells us that the 2 Sylow subgroup of its quadratic class group is cyclic. There is exactly one nontrivial ambiguous class. Working with discriminant D = −4N, this class has a reduced representative (p, 0, q), where p and q are precisely the unknown prime factors. Since 0² − 4pq = −4N, finding this form immediately gives N = pq. The coefficients themselves are the factors. Of course, finding that ambiguous class is the difficult part. Even with RSA-330's unusually favorable index, Shanks's original algorithm would still require an astronomical number of operations. This is not a practical RSA break, and the index does not measure difficulty for NFS. But I think this makes a fascinating idea for autoresearch. Imagine an autonomous system exploring binary quadratic forms, alternative discriminants, class groups and methods for extracting 2 torsion. It could generate conjectures, test algorithms on smaller semiprimes, benchmark computational complexity and investigate whether we can find useful ambiguous classes without computing the entire class number. We already have a public target, a measurable difficulty index, a favorable algebraic structure and a precise research objective: find the unique nontrivial ambiguous class of RSA-330 without computing its full class number. What fascinates me most is the underlying idea: a number can have a measurable mathematical fingerprint of its factoring difficulty even when its prime factors are completely unknown. At least for a particular algorithm, we can predict something about the work before doing the factorization. Shanks was exploring this more than 50 years ago. Maybe it's time to revisit it with modern computation and autoresearch.