curve #3706

y² + xy + y = x³ − x² − 1743521616453919485537895026371913758882959704230x + 886101964879244232952853377916898623568789978562234327233568452261401397

a-invariants: [1, -1, 1, -1743521616453919485537895026371913758882959704230, 886101964879244232952853377916898623568789978562234327233568452261401397]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/4ℤ
conductor (N): 1097108044453769307704470853301745036152608455549132667550
discriminant (Δ): 8463795652798219847254681891108624528523966691610296553193455247881185844592110297436339102837526056208968044593576885433223519507988480000000
Faltings height: 26.3722
naive height: 344.8536
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 43, 47, 71, 109, 139, 167, 181, 227, 271, 523, 947, 1061, 1063, 11953, 4943153, 588848633
regulator: 70420748372785316.443434931413649388959534356512950258128009794
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:46:54 UTC
last updated: 2026-09-30 01:46:54 UTC

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

(905066969961146008229129, 222441735759337282248602378541794235)
(-13888899672223282560079757638599/26988025, 179934080509786911176943944383110133530907182889/140202789875)
(727483603247027421112825438673/1042441, 102254721813830093044435877500267318432223175/1064332261)
(1068131382153415551148509216764801/1414737769, 562937900048080698056344424901360514802588458695/53212531705397)
(7913300012304865951314863738061881/11866834225, 182720288847509518232902139659700947332162375052941/1292713586300375)
(464829585599352290647170121465755069/611509924081, 178252331348192822220981423068380873459189019026425/478195257042025271)
(55496958971043937342084287540522714681/72567619307569, 935672508878071963040682912384322466577080236009280525/618179093593473660247)
(529196777234928985452908403912450122021/698837933704849, 128238192156363742228008026084888851475545958452209792645/18474160246485685048007)
(1057036802833109188064960624213910048209/1426164182338921, 1707167224864848309119334816619586598493560677585573507415/53858504192550410268469)
(1092660551611250536222696470133445306009/1480726946446225, 2084889108516421849910983549614093643025346117257521231197/56978691255544223938375)
(39504709494583295130678050781909350270769/47300712177078361, 1150513374749910747897571595228124719442235039936192251450815/10287302713147817132886109)
(1889621929283145195812902630097679504941649/2438478818704569481, 71475546751301379555913139853042706241338229279738318374224135/3807838155702513328889203621)

Commentary

Specialization at t = 469252/945061 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:46:54 UTC · history