curve #3598

y² + xy = x³ − x² − 10161691619227166915749752601275x + 12464279528163644972439987454588772067704871561

a-invariants: [1, -1, 0, -10161691619227166915749752601275, 12464279528163644972439987454588772067704871561]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 3482515236760862602092865033854708270 ☆ record for ℤ/4ℤ torsion, rank ≥ 11
discriminant (Δ): 40176345626093609450376048624803082963114812350662412021098091913921373931715169100311518700
Faltings height: 16.5480
naive height: 225.8021
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 59, 79, 113, 173, 181, 251, 1231, 330217, 25236637
regulator: 5645080120695.40741510016139005863307539396397797285344993750
submitted by: dujella
submitted at: 2026-09-29 08:36:14 UTC
last updated: 2026-09-29 08:36:14 UTC

Witness: 11 independent points log in to add more points →

(1020468590725723, 56189532674193978957666)
(-825906706757175, -142455325026766108707894)
(909643946299815, 63035267962225695575511)
(3915571700257729653/1936, 1159628794065718672206509169/85184)
(3939133353161269188/529, 7194876276593453311803239025/12167)
(380898756026071861183/11236, 7402114018410477358552707047591/1191016)
(49232485319772499/25, 1186416245486369941688583/125)
(470677769767844227/324, 161625538160579580674756237/5832)
(-1072940074906780955/729, -3063982516357210035768418418/19683)
(498179707901428298100/223729, 3135133428571551444517184488527/105823817)
(-52827570631332100607208/22118209, -15813320896239343788958108361228253/104021936927)

Commentary

It is the fiber at t = 35033/156392 of Elkies' Z/4Z fibration E1 of the singular K3 surface of discriminant −163. The fiber also lies on the rank-5 subfamily, found by Dujella-Peral, defined by the condition 199664t² + 104728t − 122161 = □, parametrized by t = (−u² − 330u + 94936)/(u² + 199664), at u = 7052/65. The curve was found with the assistance of Claude, based on our old PARI/GP program.

last edited by dujella at 2026-09-29 08:36:14 UTC · history