curve #3845

y² + xy = x³ − 421406774965967963069585603310038116733186236957137864114786695x + 3329896781397420269474143720108844845817406288637117577630017209595524115341966347568817100937

a-invariants: [1, 0, 0, -421406774965967963069585603310038116733186236957137864114786695, 3329896781397420269474143720108844845817406288637117577630017209595524115341966347568817100937]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 749490559798072036091888986784838118687336415779244727226504777524062355568797721372985894022777126702369438148215075151572621054837653121996068490
discriminant (Δ): -670286316914471016250062857822933823530713082231126339493337192851016315151274041954290201107672090021905102205440592278509805926025282910729163984251454746163081053102051759467520000000
Faltings height: 34.7021
naive height: 444.2099
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 193, 3989, 2438503, 31256192162509, 1766493310800019, 69998768394560870859916128620240416438009711, 155708592026621738262889507827253639483590825396113
regulator: 34426938960317018889825430710586339445.39263387071258521956184173523753542627251832550765033834817332
submitted by: wgxli
submitted at: 2026-10-03 19:52:05 UTC
last updated: 2026-10-03 19:52:05 UTC

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

(11721108169971897557180165765614, 916258153786013576391806358225846317233748893)
(11724361296396180193671930091414, 899887709675889404122136575047777153649649893)
(11736495625156902939668286241564, 839921637235502627218705699861481216514187093)
(11751443446372011980332018062814, 768897495600260992470405827029563055271902093)
(11789584053849865739721223334314, 609161335719944927116218287503173319636979593)
(11794882176231965944317128323414, 590450785884935841554652140874019473105550693)
(11871446231301025521081236488678, 496484529788031465664075343578096208032483141)
(11891491287472087250647961769454, 537229991713329248153075614491864354785399413)
(11101145775198925705618690400104, 4455704035089235666631553102226427020682490283)
(11627589724051087094950594639414, 1418317833087603378892761560467415275716713893)
(11648850407959393454237459260534, 1300535884982076647044115628011105201934682213)
(12006772150316662034806879147024, 1043490326440402510567270777055815693427126173)
(12034993845906313634336855297014, 1195954361893268533460325100089896623291832293)
(12095985748756153965391524904624, 1537804372211516241285371322655898220695522653)
(12122178338031255610899041037814, 1687886229541719592726529309550414640536877093)
(12146333722460112763886309299414, 1827470788669098400385290765513355847298606693)
(12219869304665770357612447245814, 2257336098888633132609383664369932358219565093)
(12566584313542982173921497766774, 4330815260156885951801852117685093331811579173)
(12617536691593087675812009474814, 4639110031825791742919437140707470466602219093)
(3184780305585646120588021247782, 44945650012463934414598061309309416550340097045)
(4191864746629010269794742892314, 40460782162208986189522298776996489413244361593)
(8202441391669623012777043283214, 20620209306754890379245658182915764221983074493)
(14201611308350273540138541320374, 14474266263950769272496412196920202700584159653)
(25302214844902688941718932433314, 94158921492726422542388279460997619666129440093)
(57887643490802402368855864048984, 415831630460896115677386929251758237154667475853)
(66655667318744454521573062146814, 520951446397071758695753388352120931342488811093)
(3237575778935520723502251063440806/289, 18995029770873913923141547474310375556333607890629/4913)
(61689941162148256583431907157228046/289, 15253831323836650965662544570666226904729297018860709/4913)
(9003150263276625625323713296431526/2809, 6678203819867059072564571882016568286208477078470561/148877)
(-10513624071449481622666510667514786/1681, 5212944369497093135994577978784455236578484636589293/68921)
(512892177060206991150371826466052014/39601, 52598909518053018098616083140659247717062880588029307/7880599)

Commentary

Specialization at T = 371329/82037 of the rank-16 fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 2896392T^2 + 606968T - 327316, p = (-2T + 7)(349613T - 45625), q = 9315(85T - 37), D = -340(2T - 7)(99857043T^3 - 472240596T^2 + 110417225T - 3352240), E = 3726(72710352T^4 - 433357609T^3 + 991656773T^2 - 578072791T + 75309675).

last edited by wgxli at 2026-10-04 18:09:29 UTC · history