curve #622

y² = x³ + x² − 3105168035620338064790028998277589991900x + 66668424497574957307248919422934192234755615528554466090000

a-invariants: [0, 1, 0, -3105168035620338064790028998277589991900, 66668424497574957307248919422934192234755615528554466090000]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 670806436169815337953893530982465615029341453029546883473693424304045096438291347454167080 ★ record for rank ≥ 25
discriminant (Δ): -3925700723343610695169940396792012020966727209480842917766243985576168300187232957584498268833876991667937989452000000
Faltings height: 21.4757
naive height: 284.4173
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 37, 59, 73, 19154539, 62305123360551900720316225811339538650125523947378869162263038489867663
regulator: 14784569155374697289466993.81350737748628753426545345031028811418656253275870273518726329
submitted by: wgxli
submitted at: 2026-09-07 07:15:42 UTC
last updated: 2026-09-07 07:15:42 UTC

Witness: 25 independent points log in to add more points →

(-2258963410182418192865478620496500/42483502735969, 79022919857348776232757760144079599239508573667100/276904794372373575953)
(-8685895780774958263318220589044710550/3972806579405382601, -2146015837834273804546347162120541995463756766100946365750/7918558548618150096161702651)
(-755071031384595977, -262702524613864079431472626386)
(-15678980624917247327017515882814467604707200/277894055524101279176281, -36553135988919999625935820125610154383999414784545790591375613900/146493678775550756755095175968814429)
(3963813618824858598673517271429619/25800212930881, 235012004304890357908090727492528731396411865661780/131049369359200573471)
(31197011763140035020, -12612017749513001178814945980)
(-63791577730337829332, -71838388493139240572077921716)
(471003748366731526350, 10153508176218560626577641869150)
(31645821164454398578, -9736161409472380062953020794)
(-64345768242635522057, 7553216392839765042448938066)
(-63806540817141396410, -70883730901257084856262735190)
(4137813454573785871744/25, -252699206888997214266355117684428/125)
(-1103462195983851986550323666900/47429435089, 3671174101035187736245492984118450203145849100/10329324661987687)
(300911165282440500069650134608546988/30722657378882401, 1038552222733098116775149459422258282393937073038673396/5385030487458995209516399)
(360710261921730411122197327691802700/11693450943060121, -19397791201635685052335089240724079972245447051369500/1264486063169445383147581)
(329980973760485159755414236115363573825/10074122023500353536, -321652278848378473009498220538750052297083288885986601375/31975018944686320098688630784)
(152450547203506881253339608660758642547325/3381197816551877169444, 26613238786557801206101887880403881822002575287673188501546925/196610120686795596205448566630872)
(1172826699894827720224445295148000210/17677491254552529, 918414823217479244502405181018079931090384056617260970/2350341378776428105412583)
(1187383949088391724771035114984799865493510/2548469024445167523001, 1285000777395804534356756444658286802610676290185108086744084110/128652739657982863690085925340499)
(9789058618051698549046971294730/10783368649, 30571099527650349107425000211972096942056410390/1119777350618107)
(219411882166063740956430661420865260/5462358909819529, 33167681483468039209879937563108941031740402269303820/403710782351469220298917)
(407196382197390488427086893600/32844361, -259837295272002683903816335126808556902568900/188231032891)
(16393011990553017455886602976699055809/420795472656250000, 18958155142495053065434138606338279842293495486235362377/272964763168701171875000000)
(566761187469332322654323158093089420/11390703544563121, 228566245697387123346371657126473285509883472204467620/1215698339578735365552919)
(821755264539531331272237932907850/20030520949209, 8180239702089606088872537683578337324407067313350/89647537942669492323)

Commentary

Specialization at T = 169/28 of the rank-16 fibration of X948 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 8164T^2 + 8952744T - 964149, p = 16016T^2 - 418245T + 5035456, q = 9722791T - 1496599, D = 7(1063105316T^4 - 58094740197T^3 + 937133647457T^2 - 618259297572T - 6063714736384), E = -13(55614319254T^3 - 272746794787T^2 - 717655885809T - 2521269685298).

last edited by wgxli at 2026-09-17 17:34:24 UTC · history