curve #3768

y² + xy + y = x³ − x² − 16094645373155408352765324544938597102730665595808x + 25001376588434805852665796381140559399058067952333651791746764919407838531

a-invariants

[1, -1, 1, -16094645373155408352765324544938597102730665595808, 25001376588434805852665796381140559399058067952333651791746764919407838531]

rank (lower bound)

≥ 29

torsion subgroup

trivial

conductor (N)

266902506958654905922856094066923140913085914799417197358831930889964243688309867751416279140258180440261100097052478161793670

discriminant (Δ)

-3206153289963030366980441306829104839296869685371718576928982894382028238130772828618851764421449343721449004015863126701059115894654705451918950400

Faltings height

27.1374

naive height

351.5333

primes of bad reduction

2, 3, 5, 7, 19, 23, 61, 107, 173, 191, 8380789, 18129592451, 364247882807, 39088256028994955426458566701, 2077884029206885312031192007423544698939726702827641669

regulator

4766996373200321125541060894330491.32770772471020213016662316133333364610313983280307398793224693

submitted by

Ritik Jain

submitted at

2026-10-01 19:42:21 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-01 20:03:38 UTC

Witness: 29 independent points log in to add more points →

(7542413823898389770744863285/3481, -112640312219736798189071160197616264511037/205379)
(1533000695179357316257885, -1982666230339257601682240937148462783)
(3575599060525475782788605, -3628647825378104856009696080483854463)
(2316896431174987089551005, -385838341291068066361108814453792863)
(45673041461306011581051349/25, -162756153502021586665986378512268524003/125)
(17079500678299438247138495141/7225, -249309347879042490330032152431522049805891/614125)
(2592579898743894637023565, -837061470989127774380833881222441583)
(2911331235622785360014833, -1679445852058188583575257124597679915)
(55341372883166563357347065, -410641606088393351407804038866426674883)
(2440290388752889473889543, -507680378451275659389681044763017599)
(2810843155703222581155473, -1403521541217900331639684966470285835)
(2192215513874108239921603, 503799612054541795544816882727089075)
(-170363623990812916575079343/121, 8915808684562878013602753363779326713991/1331)
(3374811808346951624324083175317/573049, 5030408362320051569672372922370037712773523533/433798093)
(443289033120988283798572325/121, 5190632571951724232418052142490112685347/1331)
(-3559052775036584468457815, -6099266967123122179560393554335136083)
(637054573254502667445540585618614683814972509/342311774176220233009, 7741910682840391438303953348003568962936878918889674750652096948313/6333339310740126187816847826473)
(-1520272888569453004808927602032016734924851949115/1620195385635070142268304, 12924735629146824598700460500494644569886745289047544505792321159673952039/2062296413157586971573986306870262592)
(2159160678416800614487207361558854381/1309008286161, 2568650757059723746169362098050157340184955284100157319/1497661251354237159)
(120246311699150178558167977910567210560810015221183890631415211740328582078420692170997/44584231878993276341820770864943710665246390525427377338293209, 327724473850942045689146794622323521716142384804599166613395797632640864995757284990248805629061975634713869611276657011230430237/297695269220918850737412559920717228771875895521277346577463320619040167389352786177091755677)
(-439135474945047369290153946287103522284458932615733/95356648166819604579704089, 1122988875418101466753207516169172831011493997090859519020955663864380617127/931164618038770448338311001362947709437)
(1253231403087781148503348050905656155764789175253563937/520098274903389098344159486561, 171998712393309778014344480472297339435541551730513372154125846303103914558988491/375083638231515936789079003167284220539070191)
(194591118500062553372633217127783504674566405519294992820168437/146526129658215427543869861285862822009, 4333492065822938398231558681248597795583498922787857686957744249208059104333412940788944844627/1773669171198941440497062066275217928179557290960343199027)
(48672963974821420045951715917012191141717017296414218272518323557229986268551185801705796935632005365/7256420764696008294754392077496628572275219548298392540591717495067346858649, 9144003162422922920370656649278815227061332856934326089445089201991068083951835810825435901356437278797408806972283449831185910058344969447166670414979/618135499925360243872304252090572638822154717648083656397913453035290808662382853593547239821921373694685101223107)
(4477974116537898217247810867620454086578125153305940351715346782938947998397/1853542631817128179836247529485830766155175244081089, 37335863314498378560613480032731058047897132121966426292950224295432014263431631563648183882780512943852973837345/79800179639373669130096650231465925342608840855244075118477727613262168785313)
(95107038562282240070453534824425093507761276179700892796585/106461763548644653571713957599663601, 116956770093254175071340698696963893834789743706852208426971940851167656627379428729332433/34736854058626503455696107695421149957698370361675849)
(104526075320420463388764577707774012631219725291472418745811010778364731488903956932612468066993868914035577220878331576421803073874188033398274687554348697026899583528892081077115435/44781000353685164296343484198699739423640457649298482571193280407248962004918354892189423281061792401099351963093535658348121904477854176250898590001202821961, 116489699774047863958814955757505417693573125576028742393930402881814443023926384214839849850031620481853996243970388506999851659165453680572236913221652532441542673444278561489021752688824757722215128294696165745471174774712254696947272842556933789396332624660927361035697/299668218800919687088507299758270018701424011645877441076292434006082930433527032414782950053482275998345503622686995994245854871441989577328312147672192730356574295839768251824369975967176300176672869648950051385942842597999586822831291)
(595772182551337330754025027395436668801659964856362444892310829/521431891512979944433878875446246474225, 33895210235237529390900468283614996330771149141304069604821669218048802383420348130823071358153/11906836348077992453111452537711392858133460964543728599625)
(103022072287257864393726214623834163598637960216799530527907297093134308228810877485/64566477635849153573083680733483535687802562675019717854801, 30176387662342585935681978890312760699009780774743402582063350194297620442567012563470241765426830782901545362078306853369703/16406299740566930302828990093761135698059690261441428723519745409135102824973407529077799)

Commentary

Specialization at T = 18065/427 of the rank-16 fibration of X1008 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 26893T^2 - 4132T + 249889, p = 802T + 75146, q = 95524T^2 - 98001T - 52965, D = -6(9660432T^3 - 491890783T^2 + 697221412T + 666719787), E = 105945840T^4 - 2512683768T^3 + 7169850306T^2 + 52602744T - 6846149970.

last edited by wgxli at 2026-10-02 04:46:25 UTC · history