curve #3232

y² + xy + y = x³ − x² − 5639193550250558498214703168519616632419994580x + 162877188666343316895747909975841868505516043864564707572585595141047

a-invariants: [1, -1, 1, -5639193550250558498214703168519616632419994580, 162877188666343316895747909975841868505516043864564707572585595141047]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 2206007226377691217596469738393439463302418193519281262196132127945323249850
discriminant (Δ): 16549824239309936232865005477066439493814894452573345218770862811352152676861993968665667874115368580651199732179770470257914038496000000
Faltings height: 25.0659
naive height: 327.6518
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 29, 31, 97, 107, 193, 257, 313, 433, 461, 739, 883, 1259, 1601, 1741, 1873, 5563, 27997, 42437, 50131, 150131, 554951
regulator: 2102162548555379174577.5820846043226482113453199635976038764226369073129388
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:47 UTC
last updated: 2026-09-26 20:43:47 UTC

Witness: 18 independent points log in to add more points →

(46119104876827436042269, -946925030540438163752830132480035)
(55237154756942260891755259/961, 159437273159288193676142750764620256065/29791)
(159948341940648497003925, 57904503039249926188289595378779341)
(49009919077541526833689525/1089, 17621968148410143635928074430192301421/35937)
(4744634638020120737436796174063/99780121, 1493471413000935547061335099602779634190504025/996703628669)
(189189144759593845165312785381/4264225, 1125782070390250220232441346444621496341309/8805624625)
(83787354296840858178461420857749/1865289721, -36399726403843624892261635288183364830886757215/80559997760269)
(10501596172293296720150204485/221841, 148082306617472313180411324418949751455131/104487111)
(-37449261607646169864311826604720068680535974019/829659347495202872416129, 13633078666051764179816315513330684966460422006241161308347796493279965/755700511035083477235451469870029633)
(54951815616863300181278059/1225, -18168185518346934161852274207633690547/42875)
(102557117433116520275450636605/328329, 31970483204736117214091108180111695906263217/188132517)
(3442123851704855003608240138547441817472279829829/52330949865028096234992721, -3311780756224404487752096125218108827569355462032460132575572668550087675/378562792535368152549679558369299595481)
(-21937646463447208803111647015743051258460148515/276213104634230162632809, 1520849290559443814297510963988185910512320198696627243865304013687153/145166504359572163674440076149672373)
(-2254950677471623177856905121373694767724859878078298691/67889605623108000250437602970841, 9904912875452910422657704922997221834818846790867256893630120218707145349474023975/559377416410200263539850068570910708735983280611)
(3117957554780861709540825425263765941933/73536975633726361, 418434204956459639167628465308984864913803159786488795239/19941537838629550437697859)
(9453839801128413641591385445/321489, 864218458191973117407276893971342066653419/182284263)
(378966428571168514979412168857985454349284512509679/7376209890393581331111382561, -1875701635884667609344677366740932115251215939489308989058614065646948254755/633504784392494069434142542344566831284559)
(21528761743683957035340655765861077/484606722769, -60720916784193719587640656175973390104066342973787/337352670168243353)

Commentary

2-isogenous image of D-z2old141516171819-009, rational kernel x=-22151565406308899719131. 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 20:43:47 UTC · history