{"schema_version":"dataset-v1","id":"bsd-euler-products","version":1,"title":"Birch and Swinnerton-Dyer’s partial Euler products","description":"For the smallest-conductor elliptic curves over Q of algebraic rank 0, 1, 2 and 3 (Cremona 11a1, 37a1, 389a1, 5077a1), the logarithm of the product of N_{p}/p over good primes p ≤ X, against log log X for X up to 100,000. N_{p} counts points over F_{p}, computed exactly.","license":"CC0-1.0 (computed values)","problems":["birch-and-swinnerton-dyer-conjecture"],"statements":[{"problem":"birch-and-swinnerton-dyer-conjecture","statement":"bsd-rank-target","role":"The rank equality that this experiment originally suggested."}],"figure":{"family":"line-plot","title":"The experiment that suggested BSD","caption":"Birch and Swinnerton-Dyer computed products of N_{p}/p and saw them grow like (log X)^{r}, where r is the rank. Here the slope against log log X grows with the rank of each curve.","x_label":"log log X","y_label":"log ∏ Nₚ/p over p ≤ X","series":["curve_11a1","curve_37a1","curve_389a1","curve_5077a1"],"boundary":"An illustration of the heuristic behind the conjecture for four curves and finitely many primes. It is not evidence that BSD holds, and the products are not known to converge in this form.","statement":"bsd-rank-target"},"provenance":{"method":"Exact point counts N_p = p + 1 + Σ_x (f(x)/p) for f = 4x³ + b2x² + 2b4x + b6 over odd primes p ≤ 100,000 other than the conductor; running sum of log(N_p/p), sampled about every (log log X) step of 1/160 of the range.","software":{"numpy":"2.3.4","python":"3.12.13"},"precision":"Exact integer counts; logarithms rounded to 6 decimal places.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-20"},"data":{"series":[{"key":"curve_11a1","label":"11a1 · rank 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