curve #3647

y² + xy = x³ + x² − 12537268201897972664569190300x + 537407455719654951639069821483479971892500

a-invariants: [1, 1, 0, -12537268201897972664569190300, 537407455719654951639069821483479971892500]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 46743260634842018618863285423770046153816518554850
discriminant (Δ): 1356856638984335911586169363143761625585517147739564882342294941982018980165103687500
Faltings height: 14.9814
naive height: 205.7091
primes of bad reduction: 2, 3, 5, 7, 23, 29, 67, 139, 239, 313, 523, 811, 1193, 1571, 2953, 12113, 14281, 63709, 127691
regulator: 75814634895508970215.809191170947855331269101402728270937784893315083
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:26 UTC
last updated: 2026-09-29 19:26:26 UTC

Witness: 15 independent points log in to add more points →

(440897999982527050, 292756926324632028831015050)
(-1156305732313161536518030/55649281801, 11660354191656270884754823054507079290/13127721226137701)
(-190697020545093296473935/5365123009, 380627299501487686711611384039193335/392979165040223)
(-20787558961327550/289, 5075048129879403743065650/4913)
(17987491699959896/25, 2384806260747161132239066/125)
(505879232188055960/121, 359679812070870427081298470/1331)
(70793355310323558425/678976, 337402427062849740267540526825/559476224)
(1980728604421653934719/32615521, 18738227195017137089402547135/186267240431)
(24718566890043784790105/421398784, 534911727353849571300588745926955/8650474237952)
(20183125691306015348859/11015761, 2862128832376019620526025058878759/36561310759)
(931206326097028480084840/9692993209, 447227295702151041082823516575511490/954304260405677)
(4459398429426190656287870/62145005521, 1319563507756819821037782482840520260/15492066281324569)
(6981827892134623952260028945/29027192760721, 528891226933414061103493419472493896097510/156389487137816163769)
(10422144319273227829929328990/108975915357409, 525180122351283237983989168141151712477790/1137616253731042234577)
(915890509382186397291307323705/5589073598651449, 709528011558488185711408398294880123702139595/417839740266057382945043)

Commentary

Specialization at t = 365/111, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), 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 2026-09-29 19:26:26 UTC · history