curve #751

y² + xy = x³ − x² − 31345484910176375196540418777187943071895696555975520575563754x + 66974244086367930138572439137439687631297215201959427754054664641394923903641564905062093028

a-invariants: [1, -1, 0, -31345484910176375196540418777187943071895696555975520575563754, 66974244086367930138572439137439687631297215201959427754054664641394923903641564905062093028]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 4776282747306790940759901783105895319790830089483330984822983418622453861033779089848369035208681184134937652590303851833304653304505540834083695587786154530
discriminant (Δ): 33325852336798924212358255002781750848643162409838221542266087375833757728701499483024368663526925571970432220728879865486685765914998814352524349902888008470606818143164542512021250000
Faltings height: 34.2260
naive height: 436.4141
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 47, 79, 4057, 9823193364552540751566840483753526213729975402030237, 17096111711340300759286109066130442673614429088322018106205141110410887695273286848289
regulator: 1182039270716769539982870263942135459804.598784149452294895235148948578499500563314114265502806848931
submitted by: ranksunbounded
submitted at: 2026-09-17 11:19:23 UTC
last updated: 2026-09-17 11:19:23 UTC

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

(9226127515516928340367017433246553/529, 845178352989912056078783562757347839770437334385138/12167)
(-16407275549207377511941517357233463/3136, 1645161558026802396052659841289490878586409978963889/175616)
(3649546061663387973611125538213512/121, -216916458613727054450617658775773550298126905439306/1331)
(77786370007087035006721640203463053/2209, -21435660363088187707980744134491249564893107341553233/103823)
(10035392913851443030202441533183603/2209, 444535789174432877708597554936693030962236652698267/103823)
(-2440863437883496806585182291948847976/748225, 7506387198183436722573500426586313459144960023314591302/647214625)
(15206761942878365777220205948875757/121, -1873402451251093003838802294948546275399559801474226/1331)
(-901090326561722757171968094870953/196, -29286456721200816267919758038658287373547302526459/2744)
(147618019332920199319154132394343/4, 1773980786714016653155501235748223696174302175307/8)
(6049269480917845820787344485602523/2209, -134249218614654682526888148074491877634517806214048/103823)
(49388759465408660667824563604092, -344949807218583990922972707039657494526388771046)
(33897840939089040084059588415444352/11449, -427993998031164029331689865759519166214184548428642/1225043)
(43216672614051493535029866801647053/11449, 1912738521868356519065820069182000992719475842357347/1225043)
(200459294988700114166049453039272, -2837080461866222636966728069032434230732861238126)
(-195394512430799745146105456317268267/49729, 126185302349917189006550230659748362444623363852485423/11089567)
(982374070111157114747457770609287/169, 19794999518517166241421659927157573548798554345732/2197)
(-708497830712364851058937647725243/121, -9389103295187788224166795452547424056444610429101/1331)
(1084632373444989689166756905192809699/32761, -1114386078952576245959276681672998174728095102271305504/5929741)
(330258813473115926156107670861557249/841, -189774441566172164071312587869307007157490538127501530/24389)
(20544240709365006002321128915018815713129592988/52744217637954649, 89690790746719162209261147675224663739522874031831085894139871995319378/12113294672162542973771357)
(76847389179845474256567617225537018114795557/25755736650121, -13714061566413165325286944723836407760719782263748861108782457226/130710646812467226331)
(57577381617939967194022771487935962184181928808/19432776250629049, 903920419226849809747611779781670902757022384034038400399453710108878/2708958022472631619770157)
(53284480672114059163446376566045652965433416963679800150283/7499501179696762421010577249, 9251946259724021980130163997221075026020967261403929117022516084442891247465072900935253/649454255257591392569711665508515508519057)
(452921169693003069864515521419612311788934089471592492703853408/247827522014384498497610944292209, -15504147889103830787947766047043289475968974639790114364577038005033433650422553732678298693018/3901434334932068642205703410787948852622935048823)
(1502820358592955335887886137526433674456989987/245285307358569, 39348249775331177265420390608464396400332035723489114711725058340343/3841557181017968589003)
(38682839020808467502274059840083993832457795010556062023883672/10743644006643852951167179917241, 31309923870272280461961637320709207988757610955917601215169105933592495303843392713942515706/35214977412360773561477556148091371971729727789)
(1622604782226817007017253867800110516610117100625862506350826136/317525627268714360607901263645609, 35891456350686397690913683826864925614117436775085899773777411237279628306841167837698916019638/5658068147976682937986272969307140852995360268523)
(84726434463987859002237843534169153160632720814101288/5066434920741027201369, -23425490343282583401243094707260966302774625220443064855023850659417105632377418/360623233102173338499521911994403)
(353760134149932539707318300252162182191194924586111225953035757/92638668100853517827799983412496, -1534402627075974325560595774534261725075984347698483875104250561948100233080810471491367064601/891637763042772705907282325181217927233230891456)
(28925071446218022845504511608582872/7921, 773858783863807101981613028937022103974715188992566/704969)
(17522411996372506171065380326107606692160853/6024800975209, 9477124097771470646839701899050926260321109290434981826590324267/14788157159296325323)

Commentary

[Claude, with Levent Alpöge, and Ava Howell!] [edit wgxli]: Specialization at T = 370273/90650 of the rank-17 fibration of X1008 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 65464T^2 + 516622T + 832, p = -95116T^2 + 368714T + 13470, q = 5(T + 12)(541T + 177), D = 1867602660T^4 - 6723168082T^3 - 44284945268T^2 + 8061106994T + 1169293920, E = 346792320T^4 - 886449516T^3 + 4726029758T^2 + 3198036234T + 497468088.

last edited by wgxli at 2026-09-27 18:40:15 UTC · history