curve #3208

y² + xy = x³ − 1132671854654574989242276551620110x + 55462573949226333810810250935548161657710883204100

a-invariants: [1, 0, 0, -1132671854654574989242276551620110, 55462573949226333810810250935548161657710883204100]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 20373912393913499123029748197579404326128537369302233310
discriminant (Δ): -1235871948351333157788553973646409235009709305211957790300819362578280483547373501185848013209600000000
Faltings height: 18.2726
naive height: 242.6027
primes of bad reduction: 2, 3, 5, 11, 13, 23, 41, 103, 179, 191, 197, 199, 227, 311, 479, 1069, 1193, 1303, 2879, 7019, 27539
regulator: 25967003120194962.348422958752708158897845160095334827961321517828940
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:44 UTC
last updated: 2026-09-26 20:42:44 UTC

Witness: 15 independent points log in to add more points →

(-44399520153394180, 4269322522045548266957090)
(-16104057183193780, -8338270165304701692657310)
(-336200071819985120/9, -182418505127661725175658570/27)
(492042086877295196/25, 798374836975700144709964954/125)
(-47593132835631680, 1251587532590658638294590)
(-1278875587644430514420/27889, 15431653625213912567737466704670/4657463)
(9076943795353383980/169, 26868635567219558879763609530/2197)
(28243904918565163749020/866761, 5883235846913944976698254990991190/806954491)
(-1866826522319934986019880/200817241, -22976700017256170295285750341515594010/2845781122211)
(1910991823544030487326396/9247681, -2614858509844886006007285613664770718/28122197921)
(-80599034327317826711084455780/1734649677721, -6372022438776303296786920362366166992692710/2284639439188897981)
(119038406297655119374675319810770/320948980672401, -1294115917096991153869721932685104979544474106840/5749816715246116862649)
(906306091152487017282620/19088161, 869562334446665031457171563863303810/83396175409)
(-3547140396541630684528708020108728391280779620/75462925786059118895218342009, -45640361295304700342344614634657594360285469182638415991101730728570/20730055219623497863241384173728710603039027)
(173878499096521132422620/4932841, -84391290818273251369307699620123510/10955839861)

Commentary

2-isogenous image of D-z2old141516171819-044, rational kernel x=382942867905233439/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:45 UTC · history