curve #2160

y² + xy + y = x³ − x² − 13762876980213380704325189382219434975627x + 621146502300386490262181331007982048373280653342968556749595

a-invariants: [1, -1, 1, -13762876980213380704325189382219434975627, 621146502300386490262181331007982048373280653342968556749595]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 18331238590268621233314106535710120397922860155863660882325091066952394194
discriminant (Δ): 167346072033141183429379974632006750478031033425544727192802267569739842854291376330645576230692810897297755053609923072
Faltings height: 21.8223
naive height: 288.8820
primes of bad reduction: 2, 3, 43, 59, 71, 83, 151, 157, 163, 233, 331, 503, 617, 1061, 1097, 9109, 43607, 45589, 53831, 290761, 6923687, 322437920267
regulator: 291306530825459573589.263553812254122806167708695568174099517048569457533
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:18 UTC
last updated: 2026-09-26 14:17:18 UTC

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

(35360257945300555995, -422729516876384594460951251670)
(1795477355583361323419/25, 7016454011674617921658865466018/125)
(8042534823227187151463/121, -4470343076563974693107801183826/1331)
(-222236952471336952523477821/1651225, 399786565617662902933241130471909987138/2121824125)
(3076918214978948628065/49, -23065552301012127312354975831750/343)
(169616357746587882696013796381/2151568225, 16123424571895817878306388995737308200342716/99800492116625)
(6350120824115880105215/121, 277304117566453918626908244761514/1331)
(34051983603060618910385/289, 3916009758964012636407174268554870/4913)
(1651450848225731941906901649/2512225, 66108899232990349485868492316969219860662/3981876625)
(585876868897391207539019/3025, -380147686893069543468360556399554282/166375)
(443443160018110195141043/6889, 25232260468558752555555842967341070/571787)
(104985009666244650384291029/1585081, -23725863743404280556855679074238879950/1995616979)
(442195557691070119166753189667/217474009, 293571304396435706625853871949608518202109742/3207089210723)
(6290714661454804173401991011/87403801, -47781477300792823777384499957558203111806/817138135549)
(352717977739600905389025729791/5061326449, -7837980165796659907752507783234179288319210/360077947561207)
(2219104709823743305531313609628741275/44857193611738561, -2353396242240633041825819901056218117061080779680486214/9500536901863888910811809)
(175282208143880501962582013156075/2476660735081, -155720487632259008366774888759560572060772310166/3897622541887108021)

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=2/5, t=74/4903 (the 2-isogenous partner of that fibre). Found by a Mestre-Nagao sieve over t=a/b with |a|,b<=30000, screened with PARI ellrank (2-descent): 17 independent points, 2-Selmer upper bound 17.

last edited by Steps Unbounded at 2026-09-26 14:17:18 UTC · history