curve #5016

y² + xy = x³ − 21012440229715677630359801760163601035x + 37061905277635283179442063179515624769037661119331818225

a-invariants: [1, 0, 0, -21012440229715677630359801760163601035, 37061905277635283179442063179515624769037661119331818225]
c-invariants: [1008597131026352526257270484487852849681, -32021486159876884668550838283641028589834444933834470220921]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 1253778194392952346874803106376685813163046898570
discriminant (Δ): 369319790027770947389635016402517429873487513014697810840017161750894603463083272358993052421326499190976000000
Faltings height: 20.1848
naive height: 269.4281
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 29, 41, 47, 53, 83, 97, 113, 233, 251, 431, 557, 3070213, 15628708024133
regulator: 92586911930861.931281079595667542130210669681201877284342150086
submitted by: Maksym Voznyy
submitted at: 2026-10-21 01:47:23 UTC
last updated: 2026-10-21 01:47:23 UTC

Witness: 12 independent points log in to add more points →

(150773965347806541322531171452338609665/28848664420900753984, 1296409421854075601978118441423265856175078288723851999075/154948925074614657704697563648)
(3211481917650412830952933463894639630/1101588122931256489, -881655599661423567357748698148417382473961943498902645/1156189090629032178925466987)
(228580889049118734787113079311034574/63193957185750889, -45993622023156200232985804663113834847968532147846741/15885954978281247897699787)
(129105278418501535508189304575723424908890789154/43086980963626845458154375625, -8957457529274689463374280949868820159183910455550729285813566900020717/8943738593958639567959030410391879519546875)
(172452568303635708545/64, -42529413805165439919417121115/512)
(121375445961038657436215774495168301406503015465713335471370410195310/5844475241621925512743384126917266869893510580081, -1307052292709000069938854068345640066802040939391426141243711386571226324062784184886158342484353154215/14129223169673898391580405676062537841380565476425962252900163559742830729)
(88435304243577412570254136130/2143412209, 26143123238526532627413850981689710803271345/99233555040073)
(14114281438008397983462409745/5174788096, -75428717865701905784931629869579193064185/372253556473856)
(7689427515612388215887316530/2270808409, 235517859736808753661730822200297054476705/108210833114077)
(384733269805476932153059670/25999801, 7220433413984267661907285120748844396835/132572985299)
(2474069151750695983106464411070/856126423441, -544730628978347356613991603375158901074085785/792148951943677511)
(44334710079700369043525087570/9533374321, -5881151788970871932611309282756801478552365/930829135328119)

Commentary

t = 4675/8693 for y^2 + (7t^2+309) x y - 91 (t^2-1)(t^2-25)(t^2-121) y = x^3 from https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/3-r12-z3k3.ec

last edited by Maksym Voznyy at 2026-10-21 01:47:23 UTC · history