curve #4648

y² + xy = x³ − 313888858581338684446615274245582656x + 67666499744368094347876262894527788017483353154949120

a-invariants: [1, 0, 0, -313888858581338684446615274245582656, 67666499744368094347876262894527788017483353154949120]
c-invariants: [15066665211904256853437533163787967489, -58463855779134033539165088958728394127261916871557990913]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 1106047551364870575551927475736261117099622365890
discriminant (Δ): 1259764479403997562246662608225222522097850368584823467951971228293094692329002518163555103650903962419200
Faltings height: 19.1346
naive height: 256.8166
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 43, 59, 73, 89, 109, 127, 1289, 2171633, 2237479, 58975978981919
regulator: 364714331073933.66657491437850955307879987582847132822971091211
submitted by: Maksym Voznyy
submitted at: 2026-10-20 13:15:09 UTC
last updated: 2026-10-20 13:15:09 UTC

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

(650599613942904483022737/984064, -376974044679649629115433428903192617/976191488)
(481126945684807406088/121, 10453782132639111609461975651496/1331)
(284031498749102923948272/49729, 150671071466077926748762588080050352/11089567)
(3662209634231511998145217832712535603925697799353168/714639621885563024360921823590521, -220350560825669955030638522258895966379365980615782716251871093716581820708544/19104278333001859672156393542483008547297170202931)
(450898711655277327367398304/1040643081, 3828639445462990172295468580991862521920/33570105149979)
(8004128039784513964237638968254080/1832055975165529, 710476157460054388830377153990816202388116919364160/78416618014309983829667)
(143719823200445644277864772/436684609, -28307241196913633865724889691992706132/9125398274273)
(839655732851925314128119702144/2534225205625, 25166153271883411498154115881255876329908672/4034296460464578125)
(137126491963413489539985587182760754/402809957826418321, 4141667576787812280412805430831199125288335415961950/255652649075471713776466631)
(531468856199556223177823352/1422572089, -2705876712953817052745925009200040083712/53655151480813)
(478927492685746370944944774784/1086712427025, 138421806488401393766602513089534946312604992/1132848803114346375)
(106334254766271866880736858824/323409453481, -455641638304494894200412275008580250965760/183920045509563371)

Commentary

t = 2743/5141 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 2026-10-20 13:15:09 UTC · history