curve #3608

y² + xy + y = x³ − x² − 165988165697279257823367x + 25350586146647657801599199621868159

a-invariants: [1, -1, 1, -165988165697279257823367, 25350586146647657801599199621868159]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 549866032239950391699315764116011233874518242613990
discriminant (Δ): 15066577936954046198817155090494224960826559686786591648004311552000000
Faltings height: 12.2427
naive height: 172.0122
primes of bad reduction: 2, 5, 7, 13, 29, 73, 89, 586543, 2651485605367, 158625189154131660367301
regulator: 13743313880425379877964.9863611525593534537779114009576976364904783427267
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:26 UTC
last updated: 2026-09-29 19:18:26 UTC

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

(-427204525407338756549/925802329, 1711169811438529288062172457194/28169387464483)
(-50305429171462094549/1471106025, 9932383491820156897377264805786/56424271588875)
(-10608353191848937937/354681889, 1162511038573230723163942188242/6679724015537)
(-202601865058275037/1776889, 489985859314892268957586742/2368593037)
(-75655878214689317/165649, 5174267783151998306124138/67419143)
(-20308271840434289/45369, 964500153210098341438246/9663597)
(8005580883603/289, 708116619595559370078/4913)
(67190974728046051/330625, 629128409544618073562226/190109375)
(77017645148428917/724201, 58143751485889088626940766/616295051)
(1096982426593737051/5822569, 388721555462329376047021094/14049858997)
(7388271113818239047/7540516, 18559054581501057183195958951/20706256936)
(2884262485475425083/1841449, 4746466917855211093951072038/2498846293)
(10108773481877691833/67354849, 34161904948160139535959657238/552781245743)
(11136348878435309643/27321529, 22765770469104127585382959178/142809632083)
(61743867177087034437/622801936, 1544057393180285422980343488531/15542645114816)
(23502680366205605227/86694721, 13400220258348762071314741626/807214547231)
(1339931415992813303427/6600100081, 2380004642221798065685453466286/536198730680521)

Commentary

Specialization at T = 3 of the rank-17 fibration of X1092, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #724 (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 19:18:26 UTC · history