curve #3606

y² = x³ − 11397391474447372188172x + 463777509647443032189011714671264

a-invariants: [0, 0, 0, -11397391474447372188172, 463777509647443032189011714671264]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 41722361591357616334861316339541426195047099689760
discriminant (Δ): 1835044235562837506855515930668865977031841012175476158638323200000
Faltings height: 11.5287
naive height: 163.9766
primes of bad reduction: 2, 5, 11, 13, 19, 29, 41, 53, 1523005730143047156865092716375187737249
regulator: 215341763646059855314.840139823971270961993136977035053121819702039267563
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:04 UTC
last updated: 2026-09-29 19:18:04 UTC

Witness: 17 independent points log in to add more points →

(7319335368508028, 626191935404329734347380)
(-1949685829179332/17161, 38428899118225276276460/2248091)
(-58814855750039/1024, 998711111138359809355/32768)
(5987754476050/81, 3660129879609506732/729)
(36490159309906/729, 85101814847407760740/19683)
(48516824569372/729, 3376535551140904100/19683)
(60642418468493/361, 393393096627944593115/6859)
(139230143135794/2025, 207270378805158581572/91125)
(1885064929831738/35721, 20564877378990458619820/6751269)
(2418106676294202/6889, 113942621328583088830100/571787)
(3895162420762370/18769, 215698422319516182974356/2571353)
(8963362779086652/134689, 7315303691287088454100/49430863)
(28249780747726690/1734489, 38392081611318671195127844/2284322013)
(267031976777775452/1408969, 119567346715650282502006460/1672446203)
(11573261326783665337/92698384, 28037360604527667430943050925/892500041152)
(147165954573088383708/20632736881, 57989291065487440375834510435900/2963706958323721)
(4241581762787104209458/58293756481, 62478552503434249180377499277300/14074502858529121)

Commentary

Specialization at T = 4/5 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (same T), 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 19:18:04 UTC · history