curve #4126

y² + xy + y = x³ + x² − 14934175398914658958707415306175682357195827040x + 701828131716934075850493140315617416734603564102554378446514730554497

a-invariants: [1, 1, 1, -14934175398914658958707415306175682357195827040, 701828131716934075850493140315617416734603564102554378446514730554497]
c-invariants: [716840419147903630017955934696432753145399697921, -606379505803431041534831449535837057335922614054117206223437317696819969]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 368471581383911304150906411175917477107662947926231328206990677277830533511360110
discriminant (Δ): 381739833896265748586517990364062652786485551764992185146916165910693406930207258182469427073555456809800679073539183245124904960000000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
Faltings height: 25.3172 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
naive height: 330.5735 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
primes of bad reduction: 2, 3, 5, 11, 17, 23, 29, 31, 41, 83, 151, 191, 193, 197, 199, 313, 431, 461, 509, 977, 5939, 19001, 174673, 1267939, 1912969, 4462121, 710126761
regulator: 494344115321346460035.256128944124266936914560082754047531699647461497248348166
submitted by: wgxli
submitted at: 2026-10-10 00:55:47 UTC
last updated: 2026-10-10 00:55:47 UTC

Witness: 20 independent points log in to add more points →

(72283226534917608853077, 88526640179794024294144952663541)
(68048431385610152293757, 827792594899211997586285325805621)
(68104333527294733558357, 792215790901139046218744793759221)
(68394704285612266620757, 590594921027844443504958055037621)
(72596334177508178742117, 510840197026914512163945249102581)
(47547194749894649923757, 9962009213016588235090135554510621)
(63818774199078363493577, 2944615646591403969158492323789041)
(64674931391070978000007, 2546897919783592359478864338043121)
(66390724094355113729557, 1723276843332967483437188813458421)
(80138288035995595971307, 4437184311404011047409773682145171)
(81190032490615863195157, 4950973392560920000645427479148021)
(84871859858071334964757, 6759337064900707848210614425541621)
(16577656219628052279685, 21419858910007073375349189628943093)
(-65791382565894139416363, 37411091713547240560955425483791861)
(-90149469659334109691243, 36269794258478236078264069401845621)
(107866731036819339427607, 18600587496547540213914071588804721)
(123742460424234993303507, 27360654128139076663200142894141121)
(667188122038910098526833/9, 39351780061341784117458981887946827/27)
(9564303834105631383751717/81, 17627910937345403553838921533644443309/729)
(-3548592122248311322622507/49, 12850113564169039116714815887767078803/343)

Commentary

specialization at T = 2407/335 of the rank-9 fibration of X1152 given by: y^2 + (304044T^2 - 789584T + 138145)xy = x(x + 529(509T - 342)(7T + 12)(35T + 38)(T + 87))(x + (7147T - 5893)(2005T - 922)(575T - 243)(5T + 17)).

last edited by wgxli at 2026-10-10 00:55:48 UTC · history