curve #421

y² = x³ + x² − 9174555732805735926833867064540x + 10766496995514664777353668694668185576606491888

a-invariants: [0, 1, 0, -9174555732805735926833867064540, 10766496995514664777353668694668185576606491888]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 987144298237983466032259164822995544703716502144048818566350982366800920
discriminant (Δ): -652658864320316015340756553454017077010475444260192868155587221990202463006477541094252000000
Faltings height: 16.6393
naive height: 225.5087
primes of bad reduction: 2, 3, 5, 7, 13, 19, 41, 47, 59, 179107, 1079431, 2706379548903143, 11425533467730813976378429779570109
regulator: 20269096010524485654.5177108841907270212561915768462205163476787423503888312465
submitted by: Bhavik Mehta
submitted at: 2026-08-29 17:01:56 UTC
last updated: 2026-08-30 01:39:35 UTC

Witness: 20 independent points log in to add more points →

(-328140673680594, 117225035381571512219370)
(805203246023766, 62459349091362531270690)
(-2325026738537004, -139746458427349404676500)
(1848995052737496, -11140069232738519203500)
(24998977207282709/4, 3555624557711785899638025/8)
(2287596917335956, -41833410390113824247220)
(1216644190918746, 37486422600178597022250)
(5492672777293746, -355083940047093859397250)
(-3022821025287474, 104300863858463015581110)
(-990286525866594, 137407402284836633364870)
(3218546265697716, 120742762175887836462660)
(2355876450803166, 47200147827940777874490)
(-2336018106429054, 139466196021806596474350)
(6278091626132916, -447900911415026429347260)
(915809625814071, 55968265424449980731850)
(37665240853137396, 7286958608863956701924700)
(322057473234996, 88572946696546960693500)
(16799755942525588/9, 325034754782328263565620/27)
(1965624059710446, -18091650999220094327850)
(-145162091026286774/81, 106774633120939268947976150/729)

Commentary

From Elkies's rank-17 K3 fibration (arXiv:2608.25406), specialized at t = -19/6. The 2 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers.

last edited by Bhavik Mehta at 2026-08-29 17:01:56 UTC · history