curve #3646

y² + xy = x³ − 24341212570918907364975008993335x + 46017718934663050280692506168501437161981737641

a-invariants: [1, 0, 0, -24341212570918907364975008993335, 46017718934663050280692506168501437161981737641]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 44025871847742350339136611296547024333859559593162
discriminant (Δ): 8194057231660386994483052814756513318437208168868334196703642321903499457257297607798341763072
Faltings height: 16.8663
naive height: 228.4228
primes of bad reduction: 2, 3, 7, 11, 19, 53, 59, 137, 317, 379, 419, 647, 821, 1747, 2143, 2297, 4513, 5647, 10453, 191119
regulator: 457530714498786658.75619102066173108788265258342396232814652616880289
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:16 UTC
last updated: 2026-09-29 19:26:16 UTC

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

(2679684176319766, 5743208724939292751221)
(3830205657410930, 94746009226532528783681)
(248692530822412292, 123996459263874675185619989)
(-24357366219621893729194/12369289, 12780800324711069089205146322709305/43502789413)
(-5689713812753183550106/1026169, 106637181529781606685706325392561/1039509197)
(-3474886522146359304061/770884, 171425807990884197857194899480901/676836152)
(-10893684461757643546/4489, 90627209338751588956935955319/300763)
(1450448593689441038/961, 3359585621085550111111077683/29791)
(3617391324999309902/9409, 174881828890788008423472793997/912673)
(14661523308436763374/173889, 15204303785954660322126305820773/72511713)
(71365277967895836518/19321, 216566162988367157446427039951/2685619)
(1024993157877255420854/323761, 4829985843888039178710611304829/184220009)
(1524114951491195305814/289, 59501291886636930990369390776293/4913)
(261345408258091968329750/189310081, 319493036894756548050563346446048459/2604717404479)
(62290281358323173311179254/3667392481, 472830044484806929867950462519865063259/222093621256879)

Commentary

Specialization at T = -565/1078 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (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 19:26:16 UTC · history