curve #4017

y² + xy = x³ − 114186550683242452081225183988435x − 91676085057319505694174838136545992782383333599

a-invariants: [1, 0, 0, -114186550683242452081225183988435, -91676085057319505694174838136545992782383333599]
c-invariants: [5480954432795637699898808831444881, 79208137489524044698335410956519187915765953062215]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 2050943799731838770795008718676624626173063235165279006502
discriminant (Δ): 91654319402820768998633513846140928169165800422073578783584478927601135172740270483687126071001472
Faltings height: 17.4875
naive height: 233.0598
primes of bad reduction: 2, 3, 11, 47, 137, 167, 179, 293, 307, 353, 433, 439, 1193, 1697, 2647, 4787, 5393, 10211, 20161, 26863, 34961
regulator: 1004280681999548490389979.229253961861947900108606471390095023195075731
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:46:39 UTC
last updated: 2026-10-05 04:46:39 UTC

Witness: 16 independent points log in to add more points →

(-88187849110945654484/22801, -1860809724191803949186280039059/3442951)
(12667215611016406, -703177644289230010996007)
(26119613818766527541303668/1228572601, 114664380199031056378126469747732483369/43062698237651)
(763593796062471255647174902/7493152969, 20983294424884567755731540838576242385883/648629800455547)
(124048580120012893221121/4426816, 40290592049728136753898257559608827/9314020864)
(4186017578186409739646253826/15708862225, 270614492118712172227657339498660807431019/1968870246970375)
(23105024159530183634031778/215766721, -110501985139847730091392571459809976739/3169397364769)
(43111103429384047795328269426/3878028271441, 863150390950262895617223325808178284681831/7636888612128889511)
(-365677591322770660565114/67256401, -334796437975888563475099053677400679/551569744601)
(5853613340851598414035846/26224641, 14146076854748604835262973389707602073/134296386561)
(3121488941948439398367070161555964/90320238563028049, 165649028560432046532366255482145964376059795678753/27144235468363819348325593)
(2132079436748742664475190673648/119414867699521, -2462554253249855541445513740545324048295683327/1304931163323600326431)
(-120868623567811994188833015862694/12231851024094601, -362480436468765681618805584559222217279600655193/1352814577775656428923099)
(-35093227670839723986087298291662118496/3726388669816705368025, -87642771361136775592801574094966336353507863314935191373/227474241756000402510073296596125)
(8347519659862572621548137562135776/690527246854455481, -311413793840365921368271645810597162864650053541347/573814118218196137574636371)
(6064491994805139867726242662/112766341249, -462712674903173033403502187113165207342681/37867726755802943)

Commentary

Unlisted rational-isogeny neighbor of curve#3653 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 16: 16 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 04:46:39 UTC · history