curve #3607

y² + xy + y = x³ − 7086975756605494357761588x + 7400180857028782861538992503442851406

a-invariants: [1, 0, 1, -7086975756605494357761588, 7400180857028782861538992503442851406]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 365003356408068742591136372104878505218862693766370
discriminant (Δ): -876999247192101508782630648574498083262013124456285007854728702009424487500
Faltings height: 13.1688
naive height: 183.3123
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 37, 127, 104106275822577333713490695368930382973451
regulator: 1178462672637185654537.67855178118693901583627609039728362243066022677236
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:15 UTC
last updated: 2026-09-29 19:18:15 UTC

Witness: 17 independent points log in to add more points →

(1762190092190, 619463258219501547)
(-2018954010109598650/7612081, 63912943614770938412138314773/21001731479)
(-3101939222395897675/1008016, 264285443696287806257819643/1012048064)
(-261376175144903632/157609, 239018428640917345446280809/62570773)
(-220010246871592045/76729, 43277921786586608883051561/21253933)
(-16451602102426870/24649, 13312069757042350125489366/3869893)
(7009262823518780/4489, 113028324647704553022171/300763)
(527393293585440425/262144, 152162250385804505636840811/134217728)
(14114456354877330485/1281424, 51605913668621719243349211771/1450571968)
(7549225590518575850/4566769, 4388031331386599725724867886/9759185353)
(41049523397173768895/6466849, 242911863897926615109099406449/16445197007)
(322478938600488213500/196252081, 1203624253639314496925412245298/2749295402729)
(1679734830637804290020/772450849, 32359685977695611836070827903449/21468726446257)
(1940717828396173260500/451095121, 72041964312294245608236763111503/9580809274919)
(5415594328804961978402/3080361001, 104653267696638990196127823883014/170963115916501)
(10489102945600393147682/15210535561, 3161914144058694530910994841718693/1875930561273691)
(333770574810238191299861/213245386225, 37127429798052371354442968703332961/98473520677911625)

Commentary

Specialization at T = 5/18 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:18:15 UTC · history