curve #165

y² + xy + y = x³ − 27169953542094477689663390x + 54214482321365567882254868382411020852

a-invariants: [1, 0, 1, -27169953542094477689663390, 54214482321365567882254868382411020852]
rank (lower bound): ≥ 17
conductor (N): 5055747593755543039707388407261319968257026830606998385654
naive height: 187.3061
Faltings height: 13.4481
discriminant (Δ): 13911346942980486161686871517934679432553631814179561190438456955749228983884
primes of bad reduction: 2, 3, 7, 13, 23, 43, 127, 283, 78439, 212867, 29444935757, 529852873647245252778716011
regulator: 12547921251179166.6765132651478498300270414723508308527523598212195212239
submitted by: Seewoo Lee
last updated: 2026-07-01 22:36:40

Witness: 17 independent points

(145057606823895, 1745958976309376327161)
(522683745173552301/6889, 377014652656645941021061133/571787)
(1583807608055806377/65536, 1950214090754745481698364267/16777216)
(25355780823667965/4489, 2706170730021416266520329/300763)
(356788907950971291/157609, -129887064946403282327521283/62570773)
(401598226891907958/142129, -2799121654846465741600120/53582633)
(1431224065644819/529, 8685208265999749724357/12167)
(1908207155420615124/811801, 1336094215002135384439721779/731432701)
(9576692532879, -25929211779693034199)
(1987724382604458/169, -81072829838071536279911/2197)
(31966715613507, -178470005016992136695)
(1219124517544603/9, -42535846884691295629151/27)
(732380383456113441843/141300769, 12185573817003989870271410080045/1679642241103)
(1180741943486364, -40572172937314419910964)
(1072640820364224006741/2765569, 35127112227364754330025618978517/4599141247)
(6885162389871, 13911834614960687977)
(2562767237796, -1189915762220229473)

Commentary

Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=2906/3: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:07:35 · history