curve #438

y² = x³ + x² − 26643979890902407817591906993402300x + 1665937119886856944539473116224248568755065751453424

a-invariants: [0, 1, 0, -26643979890902407817591906993402300, 1665937119886856944539473116224248568755065751453424]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 223243626275999515172429388531596142565523604425645085402859340457191045183972568
discriminant (Δ): 11585080412284289954378843599464712400986297972622037795687456741862452941298011926641465790911858892032
Faltings height: 18.6189
naive height: 249.4172
primes of bad reduction: 2, 3, 7, 11, 13, 103, 149, 647, 104991636693989279044111798626139, 1273365940972077825163424690885798801
regulator: 345013985575145571017.355626299711263409382341193242242083691497848470950549137
submitted by: Bhavik Mehta
submitted at: 2026-08-30 04:29:53 UTC
last updated: 2026-08-30 04:29:53 UTC

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

(260836510417938520, 111635142694197531933255732)
(49027981680913300, 21851481163890135765875868)
(-182076189162702770, -21932377965399980096536518)
(123698355723982258, 16213087023667172643083310)
(85739503622021314, 3433526710534895416600158)
(878842123039455640, -810577351862223436254231468)
(-183121167806670314, 20108341664486905950680274)
(-43081347721503026, 52286103242169276562600518)
(330124024976940259, 169845969711219852492914472)
(36913577359726132, 27068642330727163214644596)
(20161197019834936, 33718801752064501705108476)
(-164502807167951642, -39966330694450421407394430)
(2147244483197250606/25, -416554381327857398381656446/125)
(80221928840917747, -6691675895819757910892916)
(1261500295989278410, -1405554933139492845734242218)
(39640196021838334, -25923984331172680864803978)
(83113969652002942, 5059232193271654973748726)
(116573938674919882, 12005124766801714759274154)
(-85076315935621517, 57592778517514063581692220)
(-127041526905699341, 54776235618978018267694572)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 3/10. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 04:29:53 UTC · history