curve #6155

y² = x³ + x² − 37836866409244473120801287787625x + 78175748144987900758810212457240059275602934375

a-invariants: [0, 1, 0, -37836866409244473120801287787625, 78175748144987900758810212457240059275602934375]
c-invariants: [1816169587643734709798461813806016, -67543846397269557152629549425463670004891818136064]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 65469003729321126992574688995255164007952398763890089561456239252508767987094839680
discriminant (Δ): 826628063331304012319742840689079270569746741572582204993012072485229803904421328669265331200000
Faltings height: 17.1315
naive height: 229.7461
primes of bad reduction: 2, 3, 5, 7, 29, 79, 113185621, 18785385693544260856406235549200074583950687043565875801172935607077
regulator: 360347245077785792023793.802833179027553557155198234750118375784768695165882187840893
submitted by: apkorte
submitted at: 2026-10-29 18:22:58 UTC
last updated: 2026-10-29 18:22:58 UTC

Witness: 23 independent points log in to add more points →

(605722620133355, 235540531225904565095820)
(-6015326015826505, -296845740651651251763180)
(213516524122904615, -98620837298937137312892540)
(4627894848720203, -46778561243115659982444)
(2359894295463485, -45024802395402416780340)
(2364071940861425, 44035038788727422641800)
(-482460934745977, -310351826717231631385764)
(2261260695508955, 64646991078495721407780)
(96667771603300073/16, -16946625407125323004865091/64)
(25939783393841765/4, 2598793818950499860521305/8)
(823648574441975, -218106001215811558809900)
(12909913713821275, -1319601112241203833995100)
(20734368029062295, -2864899247091672173443380)
(7483265229495425, 462698330799452650840800)
(5684797563351083, 216323806396745224988724)
(564539977929512075/289, -531836908070858312572773900/4913)
(11610515845059863, -1097276217257071420890804)
(2454724082966855, -9382910280066652769820)
(209487258680966686715/37249, -1495308489172645952134429179780/7189057)
(285580488197865905/256, -791653297377161496214291305/4096)
(1935211840771325, -110457464329889892047700)
(22194977805110795, 3189359120901440597137620)
(-6808373208586255, -142085514264191222386680)

Commentary

Specialization at T = 113/7 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 2026-10-29 18:22:58 UTC · history