curve #617

y² = x³ − 1243709484034594774252579512251439437046672200533376017x + 548486337730283544420152286125150570880778768270835598547316636928034349688388249

a-invariants

[0, 0, 0, -1243709484034594774252579512251439437046672200533376017, 548486337730283544420152286125150570880778768270835598547316636928034349688388249]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

146806080754283912645946691250036103103192549129178367791442836064235384246697135528381871098086268955477892367096412508075044105849343251182540

naive height

385.3407

Faltings height

30.0208

discriminant (Δ)

-6839371254967154633910765617193397576584058018906148456133493364146434908984213404430923299760960704998618563596694359437647969410852514380862878376843076843750000

primes of bad reduction

2, 5, 7, 11, 13, 29, 47, 128389, 6255280790309, 581654576289009347, 1645304557481127105891923744749911839404955923731728073376005871955915153396177394214393551235759001

regulator

316737507841880277238930117276571190.1630902381778265857238654370545557924185708501401405664286

submitted by

Mundoamundo

submitted at

2026-09-06 22:22:57 UTC

later contributions

primes of bad reduction recorded · NDElkies · 2026-09-09 03:55:54 UTC

Witness: 28 independent points log in to add more points →

(401207844857415322344882783, 10680910727683592947573429565249241990125)
(182121632965897297663029753, 18111338949770003883216876617431092158485)
(337402446925518851635277408, 12933125989005824526292687622791340936875)
(617081947293211001461715088, 3999297711495889012722204932682662448515)
(540392477881127815852246233, -5848305493254827537613420974222039665525)
(315268356194990605206369283, -13701091895117930115404794988012372770625)
(582327446229394319086327408, 4659449213586234709342704579542273181875)
(-1189648235861592736430540717, 18557965654252104699378098535017509699375)
(-125324605554612660058442867, 26502553797862328439037344719945365004425)
(-830766646564130692602116217, 31754461146224947546630708835052890052875)
(1709123117486893453587846457/4, 78007731620329153113882468449006265976325/8)
(143461216441979524941418267367/49, 50879517138398268206621806971360700820741875/343)
(660055721645257199824176783, -3890706866711907273993419123430602508125)
(110547647679020084175261633, -20306357591210261969954185316791569916675)
(8166874766017502845284972097/4, -645988039023249717405215423826377712599335/8)
(97290621638820113310982725393/196, -20059009481415882167363273684161434233935025/2744)
(3122959916756044289523399683, 164688434754558829775303734955570087696575)
(5003301901917653762143035033, -345794214134641475475763481212463735619125)
(-427583215097416368034656717, 31655990074027562532816798368396357592625)
(843893375465179421500404783, 9995593951883626224452411013655627455875)
(872210622896240079481381968, 11280298793022255217478819731741946324195)
(2289682898170944279508659105175553/3916441, -35684656516339736369674400231627045758554591960625/7750636739)
(1076978113306776751434925533, -21405742632063398842822657016189959414375)
(82106949517502108266810732921593/151321, -340183276597850215616678918208494134607795644375/58863869)
(28606183441707399915811053194033/179776, -1435409700349922191245604986756023977279107466025/76225024)
(566880224467943306389063583, 5061698736678920111317173218174477419475)
(-605882163605314342714272592, 32857457462047781009907284351350817118125)
(4950161416316280396054296643607941552/4905181369, -6156393350274898105036082445543059759821498015489875625/343544187540653)

Commentary

Fiber t = 11975/7157 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, then an iterated 2-covering point search (fake 2-descent over 3000 cosets of the growing known lattice at level 0, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-06 22:22:57 UTC · history