curve #718

y² + xy = x³ − 416891163445623604369297428825x + 103908982619412549877181724313142587147100601

a-invariants: [1, 0, 0, -416891163445623604369297428825, 103908982619412549877181724313142587147100601]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 126255005317631596807342763298768589860892860901051460856426
discriminant (Δ): -27220232785317536534993615747937996682722280519405708587443003358825187573882312199774208
Faltings height: 15.8331
naive height: 216.2273
primes of bad reduction: 2, 3, 7, 17, 23, 29, 281, 283, 174649, 19088231792553041564635095085484211371021801
regulator: 46767289689605553.53181405014226901183782307889438462740462933227404195828984
submitted by: wgxli
submitted at: 2026-09-12 00:30:03 UTC
last updated: 2026-09-12 00:30:03 UTC

Witness: 19 independent points log in to add more points →

(9170296616792636995237010072169/104487697231642624, -278510771102450767525188587460323376961934781003/33775185040654347567890432)
(418734993399354, 1662117967214361893235)
(16188600698120094/25, 1283777357143039436930943/125)
(146679156820338, -6776098312165198555197)
(1082912481946524618/121, -1124057464170229938066199287/1331)
(3032363688298170, -163471783526239672424109)
(4694139962620879426170/13227769, 38982083215251947159800979623383/48109395853)
(39492227018778, 9354497061833037506643)
(732614376987474, -13845581378925280288725)
(27531582358716203202415626/51577589449, -67767398922198207335582202722538354975/11713631606994043)
(15747020315920459176711/1240996, 1973538867265279250378817724214109/1382469544)
(109941127711328719866/279841, 129325403503767980930603936739/148035889)
(2758178830224244794/7225, 385715468400374110178366583/614125)
(827995192677409338/2209, 57651918995761362684733917/103823)
(369489510274866, 561873290832243133323)
(316100852193049317584477226/1240704149161, 5213750144329470126611386766730054225999/1381981889921813909)
(79503780342918059059686/198443569, 3053582393161354517345274680132973/2795474556503)
(2176765857213919365/5776, 249242247688980060582686529/438976)
(1342773432407132895421657185050/3436724483757889, 164805512716760310535325886294564810712023437/201473103021712757696287)

Commentary

Specialization at T = 843/1421 of the rank-11 fibration of X163 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 36T^2 + 8T - 19, p = 2T + 2, q = 84T^2 - 113T, D = 493(T^3 - T), E = -T(T - 1)(1021T - 1954).

last edited by wgxli at 2026-09-17 17:36:31 UTC · history