curve #92

y² + xy + y = x³ + x² − 4437412060110743641525245114305x + 3586842216822165612930264910099076801587288127

a-invariants: [1, 1, 1, -4437412060110743641525245114305, 3586842216822165612930264910099076801587288127]
rank (lower bound): ≥ 20
conductor (N): 2876153493562761211278364526603564191699143885403233935132057708367930
naive height: 223.3165
Faltings height: 16.4257
discriminant (Δ): 34158080673335280296970759332689207611899474944643988972429089096818399452131514624000000000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 31, 79, 1049, 71889448247, 40200713707633, 491007790268548705232623905732119
regulator: 14199318900391412470.1467052372488935270092221520193246449884249061865323699427
submitted by: David Renshaw
last updated: 2026-07-01 22:29:42

Witness: 20 independent points

(2322567250548927, 76218907815416964881536)
(1159287617040927, -792075071337327918464)
(3260380495594177, 154198893197321089113536)
(1085033861856627, 7036233563582957274136)
(1057541405086527, 8766013807279112883136)
(2051321472766927, 55821856559912025117536)
(1349853674212677, -7520658488989426177214)
(-162875228134573, -65614525056065801511464)
(1973958997485927, 50190953064845966726536)
(1115609187332427, 4989362320694872550536)
(35632774335609831/25, -1569661495703560782391744/125)
(1065196583526527, -8291243444403322249664)
(1272177369204427, -776998561028702009464)
(1577638306201227, -22646461833194926710164)
(1076965981719177, 7551525239803692957286)
(336801666567727, -46157552734600369764064)
(1157870843592447, -1099527333816823674944)
(949628343705087, 15143284705038794180416)
(11740414067153143/9, 114951115455620018279872/27)
(2666485382308185, 103506731053981149881008)

Commentary

New curve of rank >= 20 and naive height 223.316, found as the specialization T = 28917/10 (t = 28917/20) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 20 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 19:26:11 · history