curve #5589

y² + y = x³ + 23044222392670127

a-invariants: [0, 0, 1, 0, 23044222392670127]
c-invariants: [0, -19910208147266989944]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 6422206327230105208457479480347
discriminant (Δ): -229407632214986588151309624517475187
Faltings height: 5.4639
naive height: 88.8755
primes of bad reduction: 3, 7, 69672630060983
regulator: 126155696643774763.945597970601348354138215830131242159519157
submitted by: The Wren
submitted at: 2026-10-23 21:21:53 UTC
last updated: 2026-10-23 21:21:53 UTC

Witness: 11 independent points log in to add more points →

(-2543412696648263/9117194256, 31794243696073130851315/870546176339904)
(-29579061886967/320123664, 854464460153375656187/5727652596288)
(1013943136594705/2883045636, 39932956816882581189509/154802252379384)
(1838103935972078881/1502394421284, 2507670868403086068011254181/1841517894845067048)
(9518650483950069793/1227402525924, 29367961276994234606867452435/1359817165225732968)
(34007615772667965241/73234400444100, 219958278124045469825194453711/626718761024479011000)
(65194598957723857188457/28335885769336896, 16662029424037660902744315919364021/4769855138352169217170944)
(1589426003933921897015833/4070921255649417744, 2360085624903562004331840635757748387/8213704090523116337577198528)
(655481986898790803675827129/346754687591193299856, 16810501959296312474912105372143241700467/6457040511350748633090608354496)
(200404010329832031800962194025/295277858144367236334144, 92961564040250090181253369939128806486777419/160452438943909736099000829349513728)
(2097191413686528077434292443071262993/475943686249541178764435720004, 3037495105636901122386780149068296145759660081464519397/328347228548343478782038504420467742850910008)

Commentary

Rational 3-isogeny partner of jesper-petersen’s curve #5585: https://elliptic-rank.icarm.cloud/curve/5585. The source is the specialization t = -93/13 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (-10,-32,-9,-4). Jesper-petersen describes finding it by varying the Burkhardt parameters using the rational parametrization of Bruin–Nasserden, followed by a sixth-power-residue Mestre–Nagao sieve and conductor-aware candidate selection. Completing the square in the source model gives v^2 = x^3 + B, where B = -3413958872988167/4. Applying the rational 3-isogeny X = (x^3 + 4B)/x^2, V = v(x^3 - 8B)/x^3, and writing Y = V - 1/2 gives Y^2 + Y = X^3 + 23044222392670127. The 11 submitted points are the exact images of the source’s independent witness points. Every image was checked on the resulting model. Independence modulo torsion is preserved by the isogeny, giving rank at least 11.

last edited by The Wren at 2026-10-23 21:21:53 UTC · history