curve #3562

y² + xy + y = x³ − x² − 2037619837783747966746174671909195450105x + 34753468662079960434777991387015015721532327315712767685272

a-invariants: [1, -1, 1, -2037619837783747966746174671909195450105, 34753468662079960434777991387015015721532327315712767685272]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/4ℤ
conductor (N): 519230428950677293578730527990684406193342087025
discriminant (Δ): 19667751412515156515533055946132194417701863641621100090781307553246650636386074058157193252660629043311153784267890625 ☆ record for ℤ/4ℤ torsion, rank ≥ 12
Faltings height: 21.4881 ☆ record for ℤ/4ℤ torsion, rank ≥ 12
naive height: 283.1514 ☆ record for ℤ/4ℤ torsion, rank ≥ 12
primes of bad reduction: 3, 5, 7, 13, 19, 31, 37, 43, 47, 53, 79, 109, 113, 137, 167, 179, 263, 337, 823, 1171, 1471, 3461, 6269
regulator: 11346322096910.320355328543245082321549997822433481898433919501
submitted by: Matej Jurasić
submitted at: 2026-09-28 14:41:34 UTC
last updated: 2026-09-28 14:41:34 UTC

Witness: 12 independent points log in to add more points →

(11949639957780658865, 110049901695902506986408261411)
(29075015544461710109, 9401830844632903409389699545)
(30111367139098642734, 26452949367690453270779744795)
(145348801310425612811/4, 745822356328947562351652720385/8)
(-93052184760977170903950482116/1797844801, 3015490901205682777833779135016155778148620/76230417407201)
(-26208948381729277084714765441/518290756, 1086764415027601302895479704040484953324705/11799407351096)
(1185225132894867484361/25, 26430650062203595872125308862919/125)
(30085495366678035420026/625, 3430717478704657738976727636700026/15625)
(87435943082503718761064/25, 25852288811258803800305995373070027/125)
(396061730037875457898301/27889, 433934768444192878749620629382768835/4657463)
(29132943835075310915891459/1267876, 11533846522781835974627743155924802995/1427628376)
(103007236122716761930896744/326041, 1035301890967456455733027726836944445045/186169411)

Commentary

Fibre at t = 83497/251378 of the Z/4 fibration E1 of X163 from Elkies–Klagsbrun (arXiv:2003.00077, Sec. 9), who list this t among their rank-12 parameters. The 12 generators were computed with PARI ellrank (2-isogeny descent).

last edited by Matej Jurasić at 2026-09-28 14:41:34 UTC · history