curve #4110

y² = x³ + x² − 3298135651449029404516279751051766451755854277780x − 552782370410308186759859152351168724160371736988615356817601978856548736

a-invariants: [0, 1, 0, -3298135651449029404516279751051766451755854277780, -552782370410308186759859152351168724160371736988615356817601978856548736]
c-invariants: [158310511269553411416781428050484789684281005333456, 477603968034506273360517357768342160354092680069595365381670004046026107200]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 322603758958242647542220542271392585593818068317511565892515943705988412227992083496
discriminant (Δ): 2164066545967320003799831059294966185795791012526771695944070906751480849488913142941672562125528564216304700349541352221842002815301817544885487872
Faltings height: 26.9618
naive height: 346.7659
primes of bad reduction: 2, 3, 7, 11, 13, 37, 61, 71, 173, 211, 337, 457, 821, 863, 991, 1567, 1621, 1999, 2423, 2549, 2557, 2797, 2843, 6367, 13099, 14249, 50929, 59971, 247901
regulator: 10936340962246599755381.2578200802897532657729512660123768339512962299292081028
submitted by: wgxli
submitted at: 2026-10-09 15:25:23 UTC
last updated: 2026-10-09 15:25:23 UTC

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

(-184871553186442062429468, 225012542349618752484092678925519936)
(-197803862920588554635166, 303087866984339895308460235021489002)
(-201769128731269002166242, 323211088215453371923626592433087610)
(-248919569769124974264117, 502757241042822997974038672515973460)
(-312303414922310744501730, 668413521220945350229655132040517242)
(-358047707373572390990130, 763024560974320964174468240041411242)
(-478246362254196231807090, 956637182303476553723545569075950742)
(-485881041600570259586229, 966960304831297155634916997030081156)
(1905995336957719914374580, 291755125395541187164901678982120792)
(1914370442834798524390836, 386233336737090832577347167764299944)
(-735388091675861382476466, 1214468046000093431645453655411605802)
(2152982776371883318632336, 1525183412287058430418870680636439944)
(2339481037691685745255419, 2129711908625992478344101130797202992)
(2472880524182168800256292, 2532454952953632596689844811048323184)
(2778497012934956810233758, 3425416960677840608610810048944464890)
(-1473686790343775639184426, 1052213487573876022274074544358017838)
(8056845593317710639547181/4, 7902715038875248431961255637615211099/8)
(73026275212600403333847165/4, 620925298225676853622165220709904947939/8)
(6948350068445452699264433376, 579191955556212570459790441602665102568144)
(-2460885766150250062038910614/1681, 73525686329937471280038629348704718662926/68921)

Commentary

specialization at T = 12131/8109 of the rank-9 fibration of X1008 given by: y^2 + (105912T^2 - 91132T - 6860)xy = x(x + (5T - 257)(27T + 41)(223T + 139)(3115T + 577))(x + (707T - 6214)(37T + 26)(135T + 47)(703T + 139)).

last edited by wgxli at 2026-10-09 15:25:23 UTC · history