curve #3486

y² + xy + y = x³ + x² − 357585268017125482099980994221405145x + 78511122853996073065207742221224270468256045274555495

a-invariants: [1, 1, 1, -357585268017125482099980994221405145, 78511122853996073065207742221224270468256045274555495]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 2774700096374548868697941163846920764265826133256711902170
discriminant (Δ): 263453424019130340891552883112242274107583617046777894695027135298746204659847041181989518174296473600000000
Faltings height: 19.3695
naive height: 257.2076
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 43, 211, 349, 1279, 1321, 1381, 2549, 15767, 58601, 407521, 1204681, 19629975401
regulator: 166557639182156565147.7502449678432038175870283208048445044622454362580
submitted by: Steps Unbounded
submitted at: 2026-09-27 09:46:13 UTC
last updated: 2026-09-27 09:46:13 UTC

Witness: 16 independent points log in to add more points →

(436367904425598959, 74593788848706085046383080)
(-138559834988342037237487777/226472401, 890520691993236561938976947107962716712/3408183162649)
(-32334673496301996584137337/182007081, 906962413065040224076410674757764803448/2455457529771)
(-71981611334895918470177/3243601, 1717464128053591113598017046874331688/5841725401)
(-71742921369976977754513/113569, 8753577523888195469611813308727464/38272753)
(3000422110363073411257/10609, 1088947129672333688465647139026/1092727)
(9493418701686607673623/22201, 203731275184211567680442776639112/3307949)
(89450547725477665096623/201601, 7681756970030738599315102117291112/90518849)
(621918845310719667960887/1530169, 33601003606374885225370427383185864/1892819053)
(1616691537061762363015367/5929, 2055605182067286803658142200839524104/456533)
(471405565182214779354164047/2229445089, 11700799584440316886671469513538966528744/105267708767313)
(896492895387936362606983167/58874929, 26821926418671806397901752778116669705896/451747330217)
(2666692768900812844258389103/4839845761, 74348675494265537742377243891715430372392/336703229747009)
(6005247140560563430771247055/33497418529, 870635001210684441752615177489759996031128/6130798031433167)
(39889718302054752323142029167/97764404929, 707915935470821487790561470024285790945896/30568289782365217)
(462166352624941524318849543783/27656022601, 313995135278429243675591648771758741244244488/4599224214568901)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=11/5, t=383/1163. Found by sweeping t=a/b with the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-27 09:46:13 UTC · history