curve #411

y² + xy = x³ − 3591481020811836471011183576975460x + 78382925089462270043272708916456341534296382563600

a-invariants: [1, 0, 0, -3591481020811836471011183576975460, 78382925089462270043272708916456341534296382563600]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 697471501244923640231031960671964527448437948533000993040265343878361788745830
naive height: 243.4052
Faltings height: 18.2269
discriminant (Δ): 310678744110001122475227156478814685375156829579582715053262018879190161226305823287881164380160000000
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 59, 787, 1042021, 176952289, 430133491, 121398733147020929713530078079397101059846241
regulator: 155209831672360353574.757755050007669758851590780778648493274094317138972
submitted by: Christopher R. Hill
submitted at: 2026-08-28 22:41:11 UTC
last updated: 2026-08-28 22:41:11 UTC

(24772322644184520, 2148389943036834842395740)
(9797686156275732930/1369, 368919106183525688476916031720/50653)
(3127017958357403112/961, 243359800761923022074485291812/29791)
(15592050629753408424/25, -61293074017087733530093545492/125)
(1674203746121820045/16, -1863996940466392330412181765/64)
(-112390364223206198280/1681, 305401375028158891231668012540/68921)
(5986328471188920, -7556299814247835208928660)
(33550257927612894580086120/732189481, 62727270737791279502598168914350613460/19812315166379)
(41758887077046598429525320/815616481, 124816036374062328635704642716834162660/23293191080879)
(26107976801618252357845320/988410721, -43353659186484769900596374963822880540/31074644657519)
(35172590531306519700988920/1270138321, -18288836896342665268959226977127892940/45266459622119)
(131366953672641840165478920/1379305321, 1255040104245352540642133746995551187060/51226020316619)
(377944049651233472720128080/4725600049, -5652061305995881889785586267169124402620/324851924168407)
(12183626297450520, 6036075631261013757163740)
(424311742787502149391780/9406489, -82421836885359366200622052059940380/28849701763)
(153766157384885463753480/5841889, -20393994738618935000024898895887780/14119845713)
(79475869502840520, -17174088944055915491476260)

Found with Rank Hunter v0.8.6.4; family=Campbell 1999 C6, generic rank >=13; parameter=5/2; local status: rigorous rank lower bound >= 17.

last edited by Christopher R. Hill at 2026-08-28 22:41:11 UTC · history