{ "regulator_of_ZP_plus_ZQ": 0.1524601393683195, "hermite": { "gamma_2": 1.1547005383792517, "cap_for_index_2": 0.2254330198886279, "statement": "a rank-2 lattice of determinant D has a nonzero vector of length^2 <= gamma_2 sqrt(D); with D = Reg/m^2 an index m >= 2 forces a point of canonical height <= this cap" }, "cited_bound": { "form": "hhat(R) >= h(x(R)) - [ (1/4)h(j) + (1/6)h(Delta) + 1.946 ]", "h_j": 14.155496613885283, "h_Delta": 5.963579343618446, "constant_used": 6.478804044074395, "label": "CITED_THEOREM (Silverman-type height-difference bound; not re-proved here)", "measured_shortfall_over_120_points": 0.0, "margin": 6.478804044074395, "reading": "the constant actually needed on measured points is ~0; the one used is 6.48, so the box is generous by a factor e^6.48 in coordinate size — a misread normalisation would have to be worse than that to matter" }, "box": { "naive_height_limit": 6.704237063963023, "coordinate_limit": 816, "points_found": 27, "smallest_canonical_height": 0.32696454966708133, "depth": 6, "ten_smallest": [ { "h": 0.32696454966708133, "x": "0" }, { "h": 0.47669130359417294, "x": "1" }, { "h": 0.6865793890272475, "x": "-1" }, { "h": 0.920754956872399, "x": "-2" }, { "h": 1.3078991799913302, "x": "3" }, { "h": 1.550524144321646, "x": "4" }, { "h": 1.9067776152594222, "x": "6" }, { "h": 1.9996516067562071, "x": "-3/4" }, { "h": 2.018750784834094, "x": "5/4" }, { "h": 2.4679296387798972, "x": "1/9" } ] }, "index_proved_by_box": true, "index_from_box": "m = 1", "ell_sieve": { "2": { "classes": 3, "killed": 3, "complete": true, "witnesses": { "0:1": { "q": 23, "N": 28 }, "1:0": { "q": 23, "N": 28 }, "1:1": { "q": 29, "N": 36 } } }, "3": { "classes": 4, "killed": 4, "complete": true, "witnesses": { "0:1": { "q": 17, "N": 24 }, "1:0": { "q": 5, "N": 9 }, "1:1": { "q": 5, "N": 9 }, "1:2": { "q": 5, "N": 9 } } }, "5": { "classes": 6, "killed": 6, "complete": true, "witnesses": { "0:1": { "q": 19, "N": 15 }, "1:0": { "q": 19, "N": 15 }, "1:1": { "q": 19, "N": 15 }, "1:2": { "q": 19, "N": 15 }, "1:3": { "q": 19, "N": 15 }, "1:4": { "q": 41, "N": 45 } } }, "7": { "classes": 8, "killed": 8, "complete": true, "witnesses": { "0:1": { "q": 23, "N": 28 }, "1:0": { "q": 23, "N": 28 }, "1:1": { "q": 23, "N": 28 }, "1:2": { "q": 31, "N": 28 }, "1:3": { "q": 23, "N": 28 }, "1:4": { "q": 23, "N": 28 }, "1:5": { "q": 23, "N": 28 }, "1:6": { "q": 23, "N": 28 } } }, "11": { "classes": 12, "killed": 12, "complete": true, "witnesses": { "0:1": { "q": 103, "N": 110 }, "1:0": { "q": 103, "N": 110 }, "1:1": { "q": 103, "N": 110 }, "1:2": { "q": 103, "N": 110 }, "1:3": { "q": 103, "N": 110 }, "1:4": { "q": 103, "N": 110 }, "1:5": { "q": 373, "N": 396 }, "1:6": { "q": 103, "N": 110 }, "1:7": { "q": 103, "N": 110 }, "1:8": { "q": 103, "N": 110 }, "1:9": { "q": 103, "N": 110 }, "1:10": { "q": 103, "N": 110 } } }, "13": { "classes": 14, "killed": 14, "complete": true, "witnesses": { "0:1": { "q": 7, "N": 13 }, "1:0": { "q": 7, "N": 13 }, "1:1": { "q": 7, "N": 13 }, "1:2": { "q": 7, "N": 13 }, "1:3": { "q": 7, "N": 13 }, "1:4": { "q": 7, "N": 13 }, "1:5": { "q": 7, "N": 13 }, "1:6": { "q": 7, "N": 13 }, "1:7": { "q": 7, "N": 13 }, "1:8": { "q": 7, "N": 13 }, "1:9": { "q": 7, "N": 13 }, "1:10": { "q": 127, "N": 117 }, "1:11": { "q": 7, "N": 13 }, "1:12": { "q": 7, "N": 13 } } }, "17": { "classes": 18, "killed": 18, "complete": true, "witnesses": { "0:1": { "q": 13, "N": 17 }, "1:0": { "q": 13, "N": 17 }, "1:1": { "q": 13, "N": 17 }, "1:2": { "q": 13, "N": 17 }, "1:3": { "q": 13, "N": 17 }, "1:4": { "q": 13, "N": 17 }, "1:5": { "q": 13, "N": 17 }, "1:6": { "q": 13, "N": 17 }, "1:7": { "q": 13, "N": 17 }, "1:8": { "q": 13, "N": 17 }, "1:9": { "q": 13, "N": 17 }, "1:10": { "q": 13, "N": 17 }, "1:11": { "q": 13, "N": 17 }, "1:12": { "q": 229, "N": 238 }, "1:13": { "q": 13, "N": 17 }, "1:14": { "q": 13, "N": 17 }, "1:15": { "q": 13, "N": 17 }, "1:16": { "q": 13, "N": 17 } } }, "19": { "classes": 20, "killed": 20, "complete": true, "witnesses": { "0:1": { "q": 59, "N": 57 }, "1:0": { "q": 59, "N": 57 }, "1:1": { "q": 59, "N": 57 }, "1:2": { "q": 59, "N": 57 }, "1:3": { "q": 59, "N": 57 }, "1:4": { "q": 59, "N": 57 }, "1:5": { "q": 59, "N": 57 }, "1:6": { "q": 197, "N": 190 }, "1:7": { "q": 59, "N": 57 }, "1:8": { "q": 59, "N": 57 }, "1:9": { "q": 59, "N": 57 }, "1:10": { "q": 59, "N": 57 }, "1:11": { "q": 59, "N": 57 }, "1:12": { "q": 59, "N": 57 }, "1:13": { "q": 59, "N": 57 }, "1:14": { "q": 59, "N": 57 }, "1:15": { "q": 59, "N": 57 }, "1:16": { "q": 59, "N": 57 }, "1:17": { "q": 59, "N": 57 }, "1:18": { "q": 59, "N": 57 } } } }, "ell_sieve_complete": true, "sieve_reading": "for each ell and each projective class mod ell an explicit prime q with (aP+bQ) not in ell*E(F_q): a complete finite proof that ell does not divide the index, with no citation", "index": 1, "declared_index": 1, "eleven_adic_is_not_global": true, "independence_is_not_primitivity": true, "external_agreement": "LMFDB/Cremona record P, Q as generators — EXTERNAL, recorded as agreement, not used as a step", "agrees": true }