curve #4239

y² + xy = x³ − 28623928991398919165500467001947502452240040593445344054331428726899413x + 1636638097022027963025086119463360482368875730889253334875693440181143871885452358966522018914326964558817

a-invariants: [1, 0, 0, -28623928991398919165500467001947502452240040593445344054331428726899413, 1636638097022027963025086119463360482368875730889253334875693440181143871885452358966522018914326964558817]
c-invariants: [1373948591587148119944022416093480117707521948485376514607908578891171825, -1414055315827032160053674407216343458827631518869037061248632756456728481870313760875139796253841365715575625]
rank (lower bound): ≥ 32
torsion subgroup: trivial
conductor (N): 684122702134845434686407829277257946771048148866366653485194659017350885118315403150098511124484802043838663347310086552263276924164186394156161480007550876769156907379916323938650 ★ record for rank ≥ 32
discriminant (Δ): 343806736428493800347060892801581988656808193968887937483765773102242713759119858787189867513556977045860823768831205385484555174113403636880458519898961274370784403121223980435539384836498890836098707781376000000 ★ record for rank ≥ 32
Faltings height: 39.5098 ★ record for rank ≥ 32
naive height: 498.3114 ★ record for rank ≥ 32
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 31, 37, 41, 43, 53, 89, 2220713642109042278879558804334283, 350076611288142771705686553142744278077080907949598523263287577224658029115989670361016516064255191976556239262775289081591134547
regulator: 7166200113818662326391096057747821972225640.19551997275522415134967644764791311047991113332967012572432
submitted by: zevklagsbrun
submitted at: 2026-10-15 17:27:56 UTC
last updated: 2026-10-15 17:27:56 UTC

Witness: 32 independent points log in to add more points →

(-17285086047619198200275927219245378, 46111179066568373276607383102482144371741815219099939)
(68215181559115791286912139413731752, 1215745994536280739541523550709421744253343778830649)
(-1387344455679348370056438549185082782/9, 1318887523853880583666480399346070475437416669116157773/27)
(-15173111738907417143928794817757678, 45469318950895008100110363225734405329931508848152439)
(63927798059157219113420682804111698, 8248098441085605177795173740047151713081679803826295)
(-175920256516773927216987811704653998, 35039949672608908854097547246063300466459786974561399)
(-93349800702048191841766577329428814, 59120290116614451959806566061007034385548422315557655)
(573754124312442257185948307163664322, 417240744005751235417966893171909121015622127878014439)
(180006952777798089015773645901232682, 48133228635307144812860154882618375388961803603653359)
(160321396413246682065277325922567122, 34181028216507334116572719306351091991947961215512439)
(187998867721967795592534802279222802, 53850940566565683134619753906882886861610126637377399)
(66617012700760176909764459672180402, 5043034032676679080907370401847501570294251282527799)
(20473830338934141513040567654006703/4, 308831275829136886260134068291679983315164149145219097/8)
(-144242496519268397216179549818488068, 52576900190391597126212347647253606266517688605410359)
(127127961818398385902253516853523682, 7233068110664577625624691797727005745748746428983959)
(128297156333802580224247307136843998, 8721305327064650980334193697546159894082981290426307)
(58654760853591405256596003080713082, 12629454082754085813424972634967232920799950887988359)
(264668488801898558937265106101400402, 112252649320378317400537269027978355118572889895583799)
(48875938435594063723701413074480706, 18824833359492620836802938656259778030946627675230039)
(-46751488921809892782619440189415918, 53597244206965961195885993921166172011561612233629559)
(63212355736975402066975330539562322, 8935108690519176550367540044314374317678553998365239)
(149965180266399526372714179597820802, 26771171797667975348875052454770449653345087625477799)
(-182854634263257276680050224594247678, 27509307978199386953099786312866803703357383022035239)
(27279643140022388956147032953143322, 29598791605568521450032539172535594675573347618078239)
(152646461785987818202823388078235682, 28707194425824194229749493385297886276143005584430359)
(-155412007773607516887022658957572372, 48285514471575161902643629074973045338520277417757335)
(-136511375279649473687107598119871134, 54774008346958134705142692377896672040680545719542679)
(-22380805593714240644839255686446314, 47603089514353981823409086886326475736479031451520155)
(60968847620036686567594611958700402, 10867545979063818686920688106980677205378178408655799)
(7632386621288771404751045484484178, 37664490123215897851101937789570480710930615542924087)
(23257921216948228220425920362648882, 31360579392942811665262265279771839971273135748081559)
(-151470070563153081195648444471171118, 49971003620032674130056543677631999602019036919525559)

Commentary

Specialization of a rank 17 fibration of the known K3 surface of discriminant 1152. Discovered by Noam Elkies, Zev Klagsbrun, and Samuel Li. Specialization at T = -2868768/362209 of the elliptic fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 466548T^2 - 828066T + 32416, p = -478380T^2 - 1076788T + 103008, q = 82612T^2 - 180205T + 81622, D = -26666192028T^4 - 91335649060T^3 + 11566653808T^2 + 239872553168T - 53374213248, E = -47423161625T^4 + 118525002574T^3 - 104259187849T^2 + 76235905664T - 24372268764.

last edited by Andrii Prokhorenkov at 2026-10-15 18:25:06 UTC · history