curve #580

y² + xy = x³ − 78401823826941193225130170268733083516411908087532965400525x + 8837441005634314829554018341214966345085312631966412714662237628492798132015536064032657

a-invariants

[1, 0, 0, -78401823826941193225130170268733083516411908087532965400525, 8837441005634314829554018341214966345085312631966412714662237628492798132015536064032657]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

36510756000952665280030063659319619364796897551455221288210622984978144848004074984277874902165016599942052931713561475909811125772658104689009595610

naive height

418.5309

Faltings height

32.8101

discriminant (Δ)

-2896225167802746331060540565770577153858379519782348037750956793739995851951185769665765009596932652892398181366866253295336781067880834513914764669967932115236967075472640000000

primes of bad reduction

2, 3, 5, 13, 23, 29, 41, 47, 71, 163, 4652293, 3194081185984697, 423535392343235410744131349037310481667381742232754036814065399858715754147621219835975929742448950919787082558267

regulator

536782357398056620148758074268058215.8305617341933833831006254128829640844483064410011080326656

submitted by

Mundoamundo

submitted at

2026-09-04 18:49:50 UTC

later contributions

primes of bad reduction recorded · TristanPhillips · 2026-09-05 11:25:17 UTC

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

(163268459514641770549129810504, -19725018900694091793302267447617030574431577)
(124784585256674089032637180354, 31577630709888428082727290197561488097369023)
(-2590569180390155540992841083154/49, -38858546070689788056016705461857438687990829311/343)
(115865500797190816877813382754, 36177994051535236575035785920527352070737823)
(25434629330991418933095098754, -82823751771576805642349162805962825202714177)
(144703375757275589522326895554, -22855767456478365164070576581802159050363777)
(12215930896274438325573371166274/81, 15353931486887042244903578001428844652753824167/729)
(272944642633772525628348762982, 88159659377269668696238792348410134572099075)
(214483427505866191566451527234, -43456419508024177797921362216442727062153537)
(179252657989237128380086207054, 23310195911731298973042296552977382275557223)
(-845080908751078078917831444206/361, 651459229344887819913213247997667032255218359157/6859)
(106875386860428225202700931804, 40975402972446825939430594909831221897191973)
(920386563495838542234073961986/9, -1173544062278726599301337990459663088686913979/27)
(-27654588694962171990010330046, -104806782333730297372932709837717866296800577)
(3215595664594719923177513734420234/7921, -147728210158146269650634245106759234698442461638913/704969)
(123380196421816299402107024554, -32285986909037768629087968919402172137286777)
(-191941100773463051760125675996, -129671111504750856793986822719096556453291827)
(82794807378485191193785167124, -53979013172820068639820656019481167708005507)
(59602428641162420379453984148, -66153129279518389341601623850390921285945859)
(8851157271425311417634784207522538/43681, -325435408542078396086981018549889658378812224146081/9129329)
(426594369213456466496912299959346/1849, 4376418345900624713469378244699583933768434446061/79507)
(125360395751092294366310262154, 31289369562769415003133203388154535808866023)
(-9494072932461985308688166152527852254/177235969, -267821208496754010341487070620972478517737596849393763489/2359542455297)
(53279144442979233669744649245448646146/302516449, 116738417099494176946691279710257240551157594890353304111/5261668597457)
(16264251934128719250491023664581695874/60202081, 40112993315597448580088638173346855000594676251409628817/467107946479)
(3565993695928635552503818220642087356594/11028990361, 152297037336941427857536023407965148793125686079961885544957/1158253538721859)
(-38454729717665846894383787396, -108607073408165327667771337525948733902134977)
(5336328913304945287651253830493236/24649, -173535038980929046861581753734917222613756888850581/3869893)

Commentary

Fiber t = -19733/38399 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406): the 2-neighbour fibration that also carries wgxli's #356, #385 and #351. Found by a Mestre-Nagao sieve over t = a/b, re-scored at 2^20, followed by an iterated 'bootstrap' 2-covering point search (fake 2-descent over cosets of the growing known lattice, 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 2026-09-04 18:49:50 UTC · history