curve #3201

y² + xy = x³ − 1276452816330396429849684085608147505x + 555042815435204765513879909184107302630960204020457961

a-invariants: [1, 0, 0, -1276452816330396429849684085608147505, 555042815435204765513879909184107302630960204020457961]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 6354102106121224871907639158710170892231970450727352086832662
discriminant (Δ): 17638273190245233438550034825367779352190209051980746192715738540577179286729536390940540532029522582700032
Faltings height: 19.4350
naive height: 261.0250
primes of bad reduction: 2, 3, 7, 11, 13, 19, 43, 73, 79, 103, 127, 193, 389, 569, 773, 1013, 1543, 1823, 17093, 59341, 2236139
regulator: 327020041757641968.78496504085444527963887101009502143024346334682319
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:26 UTC
last updated: 2026-09-26 20:42:26 UTC

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

(238649557677636890906/361, 73558233921924458656981323017/6859)
(112191852604278305342/169, 33066403144196522366316656111/2197)
(6285687175612217254442/5329, 323254773975751890358669562681603/389017)
(89705408122534677738314/134689, 904134255346780971912381709502069/49430863)
(1629679733162751431760698/241081, 2053063550926971898282759075313951713/118370771)
(691210115411661104291978/83521, 569569249962493394605917022595064347/24137569)
(3141740834396478251815082/4635409, 352856097067897365896414369142295843/9980035577)
(367513583806590409629914/26569, 222076412343757472317162617701669657/4330747)
(10947248534856026953146218/361201, 36195938569236236087741217266735104019/217081801)
(157016722913234379688923274263367915562/250934199713184924049, 144705268521993711394682585261780139562189252637877889181/3975024252549100876634050766407)
(7254414405909497565297908035082/11022818323969, -200634069531027008966802719464432933038913019/36596451273131490047)
(32578991253238388305070477377639276344618712298/53353189428410587401463604401, 706613194800970595262843641524630960627157673096895933771907218336243/12323684735980821904664850986898183358666601)
(61679230743735683700930730973927930/762462913278361, -15316738231126276138189862585017682985213450292498511/21053683321831710413891)
(413096357667474290096855337332846589482/660647722244928765649, 628576098602956338137689699168873473858827685428550674941/16980673555508685448642877749607)
(42301857508751868564118860609657194812473725001242/62531020076992654518284376576169, -16575198593867131635669915679234811213396334620697560163741495984385602793/494473782902670389662206850949553938158066172053)

Commentary

2-isogenous image of D-z2old141516171819-041, rational kernel x=-5183611506357375057/4. 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 20:42:26 UTC · history