curve #602

y² + xy + y = x³ − 36480103192418743462226823387253009707343x + 2632357602349499837963691825055360859848962314696484742682558

a-invariants: [1, 0, 1, -36480103192418743462226823387253009707343, 2632357602349499837963691825055360859848962314696484742682558]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 1322983321576836290387546565476702569430136772677001665842838027055986763561121788830063975968070
discriminant (Δ): 113588901487193032681354200011395746133901153664116397262105544379734963625336365737391866981966740638730770277901117312500
Faltings height: 22.2096
naive height: 291.8064
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 47, 97, 109, 103645879, 21275384581265073964905283, 185273691476118527364621845918259349307341983387971
regulator: 9897757794489524625159216.94562781584850037223360806022020093320780950568687480752830
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:45:45 UTC
last updated: 2026-09-06 15:45:45 UTC

Witness: 23 independent points log in to add more points →

(88844935091396830099, 304262733441812919496062232730)
(15065383434148547539, -1444364976912842012256234976990)
(198788395516553472019, -1798894909501707573254224717510)
(-111149503024387660976, -2305198129368191807544420202870)
(34907310684607283689, 1183837224630546122342616967400)
(57143698914575003224, -856940334287964206302408320955)
(135124232369709490144, -412535233809408278556111229510)
(366892479570814059991/4, 1926112295394128740998128733985/8)
(239159264996052933649, 2754456798303103619310530009720)
(144253015716457077469, -609708107616874818065266893160)
(605279389542082431397, 14223352641780307610728910270120)
(69147562357130070769, 663678036192018116467942907990)
(851707294137214308469, -24277458382992449173650642093310)
(1190449089956221971031/4, -34068976873928521100103905512295/8)
(72454240541552024194, -607927990458186476112878058235)
(219878209023295808752, -2289433027214946106910757345115)
(15126708487933083769234/361, -7442351411109203175818070932434840/6859)
(-211651758904023292088, -933908220655844993989506455035)
(1117089680901939271306/9, -3481051580646410112817852689280/27)
(186187140040625954269, 1514770596266249072814125572490)
(148971038656097167549, -709858578952343091946882106920)
(3994593022217719691809, 252185893586879573098429644322310)
(561334872067602412819, -12610695743489322817930883981710)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -425/132. The 6 points beyond the seventeen generic sections come from the surface's rational quadratic sections: writing x_tau = a/c^2 and y_tau = b/c^3 for a trace of height 10, the slope of the bisection is forced to be kappa/c with kappa = -b*a^(-1) mod c^2 of degree 5, so each section's point on this fibre is obtained by one polynomial evaluation and one square test rather than by any search. Claimed lower bound: rank >= 23.

last edited by Bhavik Mehta at 2026-09-06 15:45:45 UTC · history