curve #3610

y² = x³ + x² − 212216694995838546067756420x + 1191442933837710854554825439162733665600

a-invariants: [0, 1, 0, -212216694995838546067756420, 1191442933837710854554825439162733665600]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1981462165998296003807157433092943494958225792032062040
discriminant (Δ): -1567647466686818845220732103888091845121404701119043310457329636069268380000000
Faltings height: 13.9054
naive height: 193.4751
primes of bad reduction: 2, 3, 5, 7, 29, 41, 43, 61, 107, 274845816139948483, 25718962493678706481977833
regulator: 580194262576957873402517.01302379910362778448751237126204528254311078045886
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:48 UTC
last updated: 2026-09-29 19:18:48 UTC

Witness: 18 independent points log in to add more points →

(13165160258177360, 1510563405478808924017200)
(-36304940567158658607800395/2339988208804, 97981363184181491190611551282526589705/3579484642983896408)
(-278598882936376282930390/39883684681, 384724520348477570228384617145825250/7965130783957829)
(-41142090119275/4, 382503484101339649545/8)
(32123100022763005/676, 5513659714701843298530195/17576)
(175101962029393970/22201, 9527690775963263010941610/3307949)
(3658596893888296550/9409, 6993124352937594244440149670/912673)
(11838785270940153860/1190281, 10440651800873287280699982000/1298596571)
(71175575980424753135/7890481, 73954071696393616265549893350/22164361129)
(95841741276619442645/9947716, 201273642379863921949313591415/31375096264)
(125721572199910613210/16638241, 298322394270307642854671067750/67867385039)
(280773973956180133010/21743569, 2492354507670554577925928968110/101390262247)
(3255261606633082063040/736742449, 368752142919293442065331373516200/19997400293207)
(7325661934442800589015/693742921, 206706927595368116295443192745330/18272494796219)
(212690915935316167087415/24390005929, 7594053555700119555067217090702850/3809060395949717)
(983448458652811542513140/551860294129, 11731789823009407645462110618854275800/409962112280492617)
(2852201812711710234347840/329938806409, 323749172826712402541058577734095400/189517840217748827)
(13074696596775087275776345/3123306874944, 107093194274175038524493399232821224835/5519782760406031872)

Commentary

Specialization at T = 13 of the rank-17 fibration of X1092, y^2 - (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #1393 (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 19:18:48 UTC · history