curve #6076

y² = x³ + x² − 531585374066183088449273117945x + 149780502699664481675757214209355338100613079

a-invariants: [0, 1, 0, -531585374066183088449273117945, 149780502699664481675757214209355338100613079]
c-invariants: [25516097955176788245565109661376, -129410354332510265264441964137612485509587668480]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 45002340928389264935278978544984321343300551847040480758707689509229958075325992064
discriminant (Δ): -77686281079132393158353912850807174373942130238094264800295840584477418186609150972305408
Faltings height: 15.9067
naive height: 216.9586
primes of bad reduction: 2, 3, 37, 593, 57193, 439133, 563490643, 377417299226393189022226470203914931977480653367934494393
regulator: 120238196861020554979022.9521382970321108861386314734497909359340681190137637596123
submitted by: apkorte
submitted at: 2026-10-28 23:12:41 UTC
last updated: 2026-10-28 23:12:41 UTC

Witness: 22 independent points log in to add more points →

(406709564503891, -924425031209462566956)
(448717251669831, 1263694681466497151244)
(441518855834647, -1069923676996867227036)
(442340979262042, 1090593967063921940421)
(646581113268229, 8739665552944955202348)
(306518407244755, 3954527546599460849748)
(367198550327875, 2023427293998970110948)
(343172938202743, 2787375786600550582980)
(140916076920679, 8813046907412868061548)
(439556531236515, 1022434580410704743748)
(223415293460674, 6493676862297759510393)
(437754375411939, 981359132772404107908)
(-7429940721988271/9, -137654894416670062446088/27)
(1774255270856197/4, -8976913968456693807945/8)
(3413147732864479/9, 44575584776541553141412/27)
(883449132433435450/1849, 175403967743049215740992489/79507)
(4716434081478961/25, 937146810351142262964384/125)
(472801039110643, -2033892491398117404660)
(951957629167717, -22503776591192606603196)
(22003611845029, -11751355569102556578852)
(185579962923267, -7584211775541544405596)
(4420962264312898/9, -72371645615242787571397/27)

Commentary

Specialization at T = -55/13 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2350252713284250372T^8 - 50405464050571118556T^7 - 952448555730502901313T^6 - 695049490520215525224T^5 - 2931749924460790132683T^4 - 2386411337210880522384T^3 + 2409436881372804430248T^2 + 4544024515422355802304T - 2684309556378273707952, A6 = 1416965056476304965893460864T^12 + 45023054173800546171881279808T^11 + 917608982150765091196202404680T^10 + 4283108867178449968605589545480T^9 + 53149163760305922465025229930778T^8 + 64588783908002903981500312561392T^7 - 30251100417436562854623526202130T^6 - 127076856645436090494117187739472T^5 + 56548211197412673092440132446600T^4 + 156704525825325938035598980257408T^3 - 11739117611564936890190411272416T^2 - 136493880790478548113078471588096T + 53534237645294085780370164274560. It is a 2-neighbour of the fibration given in curve#3589: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-28 23:12:41 UTC · history