curve #5972

y² + xy + y = x³ − 8528221325987577221628x + 308163977335241285581239029798542

a-invariants: [1, 0, 1, -8528221325987577221628, 308163977335241285581239029798542]
c-invariants: [409354623647403706638121, -266253676418262502677661627305897685]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1031701942448500238814623346383994187948218724925406
discriminant (Δ): -1328108484709708764957458296805164523651599807998227862760891234788
Faltings height: 11.4816
naive height: 163.1395
primes of bad reduction: 2, 3, 7, 17, 41, 47, 83, 419, 1247547913, 31236385927, 79043498212870820693
regulator: 890518180517049.33818061773355008335483680147052148956871052141577687607672
submitted by: Michael Rubinstein
submitted at: 2026-10-27 11:29:23 UTC
last updated: 2026-10-27 11:29:23 UTC

Witness: 18 independent points log in to add more points →

(-103497883303, 9064652142135801)
(-96874552981, 15006547747287153)
(-18614461330, 21458395700991558)
(8951791328, 15249207554787690)
(9100298330, 15208817368758903)
(14742622976, 13624964557530453)
(43791891497, 4321788832778481)
(43999330274, 4255334725032063)
(45646093385, 3740382402074613)
(46836927638, 3387358592189811)
(48717515561, 2883756917638245)
(60778716191, 3788028827243613)
(65308442147, 5454526304797011)
(93860695961, 18291918152311245)
(102137457860, 22419075812676201)
(120514957235, 32104884894306201)
(254696359103, 121071214476932931)
(1145206425119/121, 20110707554018285583/1331)

Commentary

The curve is the fibre at T = 4/13 from curve #4220: - A4(4/13) = −250530051225379351163799381830307 / 13⁸ - A6(4/13) = 1551614814321580263454294680252020288318239844190 / 13¹² Integral model. The denominators are 13⁸ and 13¹². Scaling x and y by 13⁴ and 13⁶ clears them, which gives y² = x³ − 250530051225379351163799381830307 x + 1551614814321580263454294680252020288318239844190. Its minimal model is the record curve: [1, 0, 1, −8528221325987577221628, 308163977335241285581239029798542] Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-27 11:40:17 UTC · history