curve #3926

y² + xy + y = x³ − 3060769098988416038057310823392647445759157311807430314666558x + 2056180093735682259313134720022451161553574068119780985146772214538736327458765049092635556

a-invariants: [1, 0, 1, -3060769098988416038057310823392647445759157311807430314666558, 2056180093735682259313134720022451161553574068119780985146772214538736327458765049092635556]
c-invariants: [146916916751443969826750919522847077396439550966756655103994761, -1776539600987629472046548398099618178957415160810230897546095463977562846250823137398693112741]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 26805408071927774799000455910266014028467217724488358819512397292588705717588554252370176594825217067259912163411527308486793136304454846114493997910510
discriminant (Δ): 8707784688317903469801008592719134688634564893993366198571104851944217598127623681717550731607834103941625253079933837488757911498174666462966371232859307753338413448391395263437500
Faltings height: 33.5923
naive height: 429.4349
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 41, 2930575238817476392722758093181379206639338654038667330971921189700570769472466042611145949111797282084965077593189111899932748482559363371253
regulator: 312866461744761389256203585318223255273.0065029677953105389314357565479161933806214749511381129991379
submitted by: wgxli
submitted at: 2026-10-05 01:17:31 UTC
last updated: 2026-10-05 01:17:31 UTC

Witness: 31 independent points log in to add more points →

(959895193053390503080494552770, 51076510829890759010238771695744165907123117)
(961143046047263581206819507395, 47362609385782252224815071292587219749111867)
(961157875895423381897179257347, 47317289886551648498457276949124970229080699)
(965016200981224294408527464126, 34142521659039925746309567279325483519716249)
(-8525147081466740407622124260, 1443008309364570346804819276445516157326587367)
(1056500901058800972779339122790, 41652276211642290511294305010086156002743992)
(1059154457054719506406804040255, 50209334596377208425814565521636066379074767)
(389758449449514609230281747325, 960431322024792606884665065429749441107405167)
(395602470908929205073513976760, 952493875153227209501834598944145696688736292)
(420330957806428603404211554815, 918644309650172524775373393591513735004436367)
(594198019214853126740017138790, 668783534971266158173991415984508377960379617)
(705855114313245065449977334640, 497392986695628863724602443476979471702827867)
(767008003877184987151772307695, 399721760375551822378750824049569991046873937)
(805792247012558294722678119677, 336211113396061611312500584667261510074047699)
(865921357387472602412613640940, 234690727255895580024195714116792999766384567)
(884410491717315874684426941890, 202419575270847257113456997868775269111699117)
(909690206853996795841155712172, 156939675688581406898748630802300213986255324)
(924664938503422759377086835485, 128792851387264325023264417691143712140645347)
(1127282688073030787674458806726, 195808274653023805768791460682104036800536184)
(1143684159352565591776389727355, 227115015000894332877500699358659052317097267)
(1271130162334009718905954417175, 468402131066459942627667046378845276816784167)
(1347095883729304334110898869640, 614460345750094416039450464327203305312186867)
(225557896375529208128735708795, 1173573610923045472972310858961564787449217617)
(1969385374036399013329254507890, 1914827805192497557014616939938782303073082992)
(-1037486474957583701270618818600, 2028536129153356696724044765686238057093145667)
(-1660933580209057099389841008610, 1599342430312300402062470978203742542490821992)
(-6830191222013220535090254210565/4, 12142775180346736682164097735643169347905452311/8)
(56507077430347299094932096514985/64, 104981092062921401579854198886731511930021361279/512)
(1489042519034866858323405638150015/1681, 13780933185694892244332692057049955255686841620007/68921)
(2750305590109547174887715573455885/9409, 994132947831202686258242925102684040135137564757991/912673)
(48269515570754580848970477422304683600/45738169, 11599824786226297766143192312562981963567101751027423649/309327236947)

Commentary

(Amusing coincidence: the ln(N) of this curve matches the primary laser-cooling transition of the BaF molecule in THz to 8 significant figures. This number has been burned into my head since undergrad.) Specialization at T = 229817/49160 of the rank-17 fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 194222T^2 - 546468T + 628841, p = 9255T^2 - 65810T + 19856, q = 561868T^2 - 824209T + 405, D = 69(50861980T^4 - 423923797T^3 + 327033605T^2 + 272380918T + 2084880), E = 23(36050220T^4 - 167366909T^3 - 3016081708T^2 + 7157890125T - 3824049600).

last edited by wgxli at 2026-10-05 01:19:34 UTC · history