curve #3723

y² + xy = x³ − 39308487537865888978574883363887176366150x + 2915709249148784249365513826706507891862806922457247216983396

a-invariants: [1, 0, 0, -39308487537865888978574883363887176366150, 2915709249148784249365513826706507891862806922457247216983396]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 100333804916069595557644850229910002358198035153406924766836550463004943183082
discriminant (Δ): 214630999897146538443497903936044513379774305041319347527909037118878825851636325164231778102127306510755521136385453473792 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
Faltings height: 22.2476 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
naive height: 292.0304 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
primes of bad reduction: 2, 3, 11, 23, 73, 83, 89, 127, 157, 197, 199, 353, 541, 563, 757, 853, 4703, 6449, 25471, 29833, 190711, 203857, 304063, 437153, 18963911
regulator: 1790925129467657586994.5116522770958647750216227929476745006770655090928278
submitted by: Matej Jurasić
submitted at: 2026-09-30 20:04:52 UTC
last updated: 2026-09-30 20:04:52 UTC

(82154123366105736668, 490752216782240291682367020794)
(86054788751272530428, 412671885829612476500750326874)
(97662831718404544490, 90785116650298612930073682344)
(97995615454675809340, 68669486734604260403481050794)
(166772223672534557228, 999288242795991486167828555498)
(223366969711397701868, 2297802375137082609837650669738)
(259270979186432696876, 3186334560784181186002200223466)
(765380088803404453904, 20523024742194269159579650132490)
(-7017289532878381440586048/69169, 44034700096401651235079674199112098458/18191447)
(-363256393447913835688516/1849, 138978223431811571494556592780402206/79507)
(47342101143524825435036/529, 4119120930736917325365395607227350/12167)
(96839480927867711184652/1681, 63257350076668278098087663228491514/68921)
(1228371761076591532074676/8649, 357659338502102555346082486808721322/804357)
(1968657592940072489317436/13225, 916128005497521663559882934890412414/1520875)
(16889647307632474139531348/52441, 58417922341971681524327209983883386722/12008989)
(59480491256774166401454232/51529, 452358815269176752046298203357019426862/11697083)
(154842705322316166418022668/3606201, 7829259946254344230289481295412715245662/6848175699)
(1295156841902090120518212764/25573249, 132822119930565708013639206369758808908458/129323920193)

Specialization at T = -8899/5648 of the rank-9 fibration of X400 (equations as in curve#802).

last edited by Matej Jurasić at 2026-09-30 20:04:52 UTC · history