curve #1932

y² + xy + y = x³ − x² − 179973799601179565094960489797x − 27199003692952159515636237908567379454872131

a-invariants: [1, -1, 1, -179973799601179565094960489797, -27199003692952159515636237908567379454872131]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 2848415887139195817481684276358617283728873972605630
discriminant (Δ): 53497569865809061764003661357827292194298253688166674793844878590920812182617673434400000
Faltings height: 15.7675
naive height: 213.7014
primes of bad reduction: 2, 3, 5, 19, 61, 167, 173, 193, 197, 211, 349, 601, 709, 2213, 52201, 175621, 1134769, 34412689
regulator: 340960512803904.6349739256383882859366766244155573364650731650186
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:17:33 UTC
last updated: 2026-09-25 19:17:33 UTC

Witness: 13 independent points log in to add more points →

(-190300819759463, -398155048562486108524)
(-226990250192873, 1399184960247620966336)
(9137875172505277/16, -480208753604198367677551/64)
(-297324884850223, 165967515262524246836)
(13608222212280697, -1586677282597752428834284)
(392574763133307589/16, -245933811538906905460117057/64)
(365678253760023964123223/726140809, -61506999404501044042179973499931632/19567316380123)
(1176395092169627, -37270635644190839533664)
(-158371674407822951213/707281, -814111030453610794235910732544/594823321)
(529729090626517, -5110035534009175739404)
(-77388991405479960579229488896596027/289238316077989653904, 6600662588447024372781423118082099129523856822587833/4919078312637627888320090454208)
(336570969930623107, -195260607725371999779087364)
(1991618953481724127, -2810666816229827256027603664)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (1,-11,45/2).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:17:34 UTC · history