curve #4002

y² + xy = x³ − 525156415498027055401492147180x + 146398741241317521385641270732051644453845136

a-invariants: [1, 0, 0, -525156415498027055401492147180, 146398741241317521385641270732051644453845136]
c-invariants: [25207507943905298659271623064641, -126488512432498376288455973770440609715556794465]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 304327139592757069203804262254572708025894
discriminant (Δ): 10400473751702908114072334499083225541361676599920073935295178625147116347912157971698432
Faltings height: 15.8289
naive height: 216.9141
primes of bad reduction: 2, 3, 7, 11, 17, 19, 41, 53, 59, 97, 113, 157, 251, 271, 373, 431, 1471, 1523, 5569, 67751
regulator: 2408388679028026634.52120128114855798751596271404478265372964386968
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:43:30 UTC
last updated: 2026-10-05 04:43:30 UTC

Witness: 14 independent points log in to add more points →

(2928819534860289849088/2461761, -134112837988792328662073772336524/3862503009)
(7039841506762152848/10201, -10935362462153182736761545116/1030301)
(349706028202300531808/654481, -2267877342822636507069853065524/529475129)
(28567689591270472364/66049, 7024209237246875820798504784/16974593)
(41946974186528608/81, 2660748946142536652972876/729)
(611013394853738, 7323610146183644355104)
(1229455280401630086954584/298149289, -1343458797094614845741325759641639156/5148143773163)
(4988242059013053551095625858/72012259201, -352287804233169507357779453172197779689886/19324561768847551)
(185135797838665877498/373321, -642219187069417426762587260576/228099131)
(204389428392956704/225, 69079974931227700039232212/3375)
(48075655317356815534/8649, 330638543577252647087275971472/804357)
(128191434702313881152/267289, 304304741057565928985496905372/138188413)
(22539475977120707244686/51825601, -191975129613409864388923013757026/373092501599)
(12970536073932458069309354/30180375625, -1498416466414872844834161583353863008/5243085755453125)

Commentary

Unlisted rational-isogeny neighbor of curve#3640 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 14: 14 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:43:30 UTC · history