curve #427

y² + xy + y = x³ − x² − 26422761132426221314307634025955x + 60338761815319965289688140627358385040885056547

a-invariants: [1, -1, 1, -26422761132426221314307634025955, 60338761815319965289688140627358385040885056547]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 88941907505430897549881539052676922760357137113333198823829960967920984550
discriminant (Δ): -392178934915463317587385364218533001535084788286412387772961994010848738438549615708708864000000
Faltings height: 17.0664
naive height: 228.9558
primes of bad reduction: 2, 3, 5, 13, 37, 59, 73, 94065547, 1014246206813831009964763804797915611922643282349422786051
regulator: 382976288406273024674.48720510987172495067877611909979688105031901971683460956387
submitted by: Bhavik Mehta
submitted at: 2026-08-30 01:39:33 UTC
last updated: 2026-08-30 01:39:33 UTC

Witness: 21 independent points log in to add more points →

(3970269019018939, -134226622675063174573870)
(-5757989916959461, 146893765090808635810930)
(62634285181819, -242247882946572957564910)
(1072849867984219, -182279904964035407808910)
(9855100990107919, -870113163592952809345810)
(83087503181307781/49, -49002387210887323746704830/343)
(-2925741177921125, 335560264944516901694066)
(10660643972903899, 995103083101124930620850)
(4933685926223899, 223761641424416324020850)
(160872432696326299, 64491693079603774024219250)
(242505353506819, -232261409413411107801910)
(6937576609184923, -459275716753607799453454)
(6779609048639899, -439106610303357658699150)
(1088327420659739, -181304228057016689513870)
(-3744970816831205, 326754864737302521984626)
(5381533202006299, -272025439503167657265550)
(5671410923746555, 304802886745993796283506)
(413099419529490499/121, -132588981603691868443259210/1331)
(-3604746457565201, -329765567531018289272050)
(-3058153940302565, 335474058501157034757746)
(3480589307627419, -102652755326944083572110)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -17. The 4 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 21.

last edited by Bhavik Mehta at 2026-08-30 01:39:34 UTC · history