curve #1925

y² + xy = x³ + 260526344376084619700653125x + 1477719779262051648182126957716549533620625

a-invariants: [1, 0, 0, 260526344376084619700653125, 1477719779262051648182126957716549533620625]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 171427577232061947088270456975440421292364577170
discriminant (Δ): -943340413990995819810054337492191784520981404953867345098072182279654926340346964480000
Faltings height: 15.3678
naive height: 207.7213
primes of bad reduction: 2, 3, 5, 11, 13, 23, 41, 43, 269, 283, 349, 463, 12791, 29101, 179743, 244303, 113982049
regulator: 15371003679745.84402822975186774131934082571792689360933507399765
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:16:11 UTC
last updated: 2026-09-25 19:16:11 UTC

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

(5584476019667550, -417327789442800665326575)
(-58305292862941665453958427701619948775586/541787672309704677001559569, 5686669873354107031033738466589421761657244310837052363741961/12610830210601839289639846606341294725753)
(11268052774694070, -1196117874547760114368095)
(-12202233466291066852570312001720345077657478423186476240073118450/588117879598703661563577583019799401036271570998521, 17253434667612393343378490631633158970948211102010895853925679821119333425265264767913055385370275/14262530105328389262981812536733882634539980635096142043648687700385224834931)
(20605208403792864557206158/63057738769, -95604971612599259978641677537138606423/15834617955499897)
(222221773767824220905386940550/567353859894481, -106124344283771835014689129625886171838884825/13513909953413882784871)
(20451123681450, 1221311369699641134825)
(-40320279417417401697615281898/404666370575041, -5536502588110723937714103952480601006255703/8140398608392479344161)
(40060086633999151723025146050/31678312465801, -8021614597424081654005019802520033896098775/178296598288578592549)
(-26598357756817677261954/502432225, -12917306612126619735567406461586401/11262018323375)
(247166244173218070/3481, -279662226044493375306590525/205379)
(261049607048584072636366/1113490161, -141117483862575071404634794353617927/37156053182409)
(-19595959995726677284402131257804825765154090550678461173736884130/189727779432349943783927274081075592660363486848409, 1543866627561163608098910444217112922700408749731305888937524200063252911013480498701948526425325/2613342827042063237932839628896869050153228953538372134994308656306508362173)

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) = (2,3,115/9).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:16:11 UTC · history