curve #429

y² + xy + y = x³ − x² − 3548379898700520547221772345212x + 2596244984555180683994653333342076110656596799

a-invariants: [1, -1, 1, -3548379898700520547221772345212, 2596244984555180683994653333342076110656596799]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 3294048358302772938894219727593961593965230803928008677606014515626270
discriminant (Δ): -52521162535382074717439043300866350950112927203157501676552632309040698338202829093120000000
Faltings height: 16.4161
naive height: 222.6639
primes of bad reduction: 2, 5, 7, 11, 13, 17, 23, 31, 1968749, 92346409, 1423034287, 104937741999596607860891696642716903961
regulator: 10087907745235037808.4720051819669974776346184561148590643753324148278639580912
submitted by: Bhavik Mehta
submitted at: 2026-08-30 01:39:35 UTC
last updated: 2026-08-30 01:39:35 UTC

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

(3208068410526357, 155657592404062027312421)
(1095482427449557, -4870896459426844043579)
(1855942811805097, -49025441849159211750179)
(-2138103036212843, -20217361031895494895579)
(1574758890175057, 30225692485868306141421)
(973548032736657, -8028164914831322574079)
(458554162517225327/361, 80716916851009337797211639/6859)
(916575819015157, 10672928533516225548421)
(1324306440467157, -14820755597628058891579)
(595473871966197, 26351964156054766354821)
(1439056872582357, 21680528321812318785221)
(-1739162520750563, 59220211184259972033421)
(50652444939397, 49159341147838724774821)
(127771885628597, 46313580597341706074821)
(887669226338657, -12078999566166863412079)
(5160109421258096573/4489, -1811451714849670882603148377/300763)
(103285830196582117/81, -8775603785644593513829091/729)
(1022925801399957, -6073044553877013864379)
(524045495469797, 29675710101367413054821)
(1030447959551157, 5829021636357064948421)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 25/13. 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 01:39:35 UTC · history