curve #62

y² = x³ − 908800736629952526116772283648363

a-invariants: [0, 0, 0, 0, -908800736629952526116772283648363]
rank (lower bound): ≥ 17
conductor (N): 29733076040369196065644941545122876271020366852499870815169972871684
naive height: 165.3025
Faltings height: 11.8329
discriminant (Δ): -356796912484430352787739298541474515252244402229998449782039674460208
primes of bad reduction: 2, 3, 2195745961, 413891567044514092637683
regulator: 3160234672289479672416233508.12027866509734322914490119491393668142907407
submitted by: David Renshaw
last updated: 2026-07-01 22:35:40

Witness: 17 independent points

(298790075644804342283980784868871/338043529350370813225, 5161349606204399690920252276261178070200995539294/6215254847769749632414964520875)
(270591370675628585666599951092583/1330170417506751082249, 4204011896288889911631034546142768651699741660430/48513312301533384558486215185307)
(325711162975741591923445133588371/1052929463779991398225, -5787321283665994825424977201858275717685219519956/34166376148636124826781558558375)
(1211712532726128720491413342757359/16223941796409657025, 42179303035772843722884231140233126307757275471898/65348341423942992849106321375)
(1283241307160188714305985428097387/3583547576733370401961, -45511555930423359204879858346721593921694733454080/214520974960756038450867149698709)
(1193485252300183998204780009216121/12129239979490109440000, 8849454084306730092832158704556981698570573221581/1335827510135299734248783872000000)
(2593675287800486498913879030036859/25807176887944349617225, -42751818514177766312939758214885318731526695289148/4145822916881649392274547955517125)
(3077300921695987166539964733710863/31522550027409402194689, -25979911623890974715859271180333902196867407684750/5596699818568151395152805066539137)
(9972014278736676784554663417348859/88394686577381806455625, 603270214553538275584451704094996369762735784335148/26280839889690220249196901117453125)
(11950777486078440569963034830397097/67332468107359410846916, 1195575217084807112146406927423399551700384025930165/17471765860802684591886166103771464)
(32897935731038035198457842129038763/208286420018823947157289, 5233788033009907864404702853919512596771275642180280/95058604246594295490906437252522837)
(20633518747541148948800944355581657/140573422665650005479396, 2502007264930004827301247855826744471532652889922755/52705365409812184960750748589475944)
(61238135983124397127054621223243257/626138814511227732809796, -2561146939700617106205336638889703164869384364340525/495456967270561665445673889123955656)
(2324524904636311085961808350770265673/9878844188981889047430144, 3418217044393581168928235873217774854469695847280879045/31049828344088581964714849238008758272)
(6916737030865166929678826270687531593/20533409506285541218903104, 17973246213956629567348995135636986789359373964277484485/93044692048178135854894591156891848192)
(1194537640939104093442656918171419153359/234178544464776436764868225, 41285568778238657684933397401332433223076982094874752397602/3583609287614365760517866287425122866625)
(28221301625148907160620412386322712252871/1214634268989054295088514025, 4740951854035394390115339947785818419239050274616667906265794/42331951983322992425954355521097926555125)

Commentary

Provenance: Noam D. Elkies, Mordell curves with large rank, and Elkies, ANTS XVI, section 4.2. This is the lower-height member E_k: y^2 = x^3 + k with k = -908800736629952526116772283648363 from a 3-isogenous rank-17 pair. The submitted rational points are dual-3-isogeny images of Elkies’s 17 independent integral points on E_{-27k}.

last edited by David Renshaw at 2026-05-29 04:08:07 · history