curve #3605

y² + xy = x³ − 26230538129595285815506115x + 52242922963576365663861406254548148417

a-invariants: [1, 0, 0, -26230538129595285815506115, 52242922963576365663861406254548148417]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 18013703274881884027646086485877363482258088098390
discriminant (Δ): -24015432858013284750880854644069508278550548303172558995669136301962240000000
Faltings height: 13.4678
naive height: 187.2211
primes of bad reduction: 2, 3, 5, 17, 19, 29, 31, 43, 89, 107, 881, 1019, 2557, 70003, 168554317, 186440364443077
regulator: 60166503614881545458.2558016767171747379225648736457691244405193977634552
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:09:14 UTC
last updated: 2026-09-29 19:09:14 UTC

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

(1138246283010394, 38401657162377707818273)
(-55907139067767097016/23145721, 1121909348831440486400689410083/111354063731)
(-43664100501439929122/7823209, 108919047245841188950637650525/21881515573)
(-123613328914271426/22201, 16762363226032609673849237/3307949)
(188555470717576/49, 987388868875899999559/343)
(234590353485166/625, 101802973065605118303889/15625)
(1033155527863279/196, 21332681436352375499747/2744)
(5707723131764494/1681, 105083923308329167742273/68921)
(29901188062769134/6241, 2980634917253163084189607/493039)
(112173600309774766/1352569, 11130628389353458171044553691/1573037747)
(115314076173532654/57121, 37429357790935070076355847/13651919)
(13423916599354134162/4765489, 8741075532634761300186356187/10403062487)
(39002962268941280686/14691889, 64635832188531325878843106481/56314010537)
(170968571998919373049/41165056, 1020941246659273560027570103013/264114999296)
(73296498463965346894/107972881, 6613714709074224972851579116223/1121946206471)
(908402938910805783214/312193561, 4107360266103865771529190966917/5516148029309)
(26418355633818949787614/3699194041, 3405889221909462084478379022777173/224988680767661)

Commentary

Specialization at T = -3/2 of the rank-17 fibration of X1092, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #1958 (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 19:09:15 UTC · history