curve #810

y² + xy = x³ − 532491543698965385143716908013986103330350x + 149381728426087005376107064469239512209181414698102849693722500

a-invariants

[1, 0, 0, -532491543698965385143716908013986103330350, 149381728426087005376107064469239512209181414698102849693722500]

rank (lower bound)

≥ 25

torsion subgroup

trivial

conductor (N)

1378397082251839370263140297286229416548691765876717448982399683310777484953385725077098223803466345145690

discriminant (Δ)

23099482376905988457966054201887058788535363928898683076895413572182222503235231396509690958038250558023605895963900774400000

Faltings height

22.7674

naive height

299.8488

primes of bad reduction

2, 3, 5, 13, 17, 19, 37, 666229087, 16936652169366103, 1489030358063875190255171533, 17601544640376275918143047501174198706494809917

regulator

417759432785830517182243520489876.4116138352584306207496146142877126157272764143668378380

submitted by

hdeping

submitted at

2026-09-21 00:10:47 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-21 01:38:41 UTC

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

(909117052465725803260, -20412354269520390940209323216510)
(288584445201706457650, 4443708777781628474716304792200)
(-1955306005496150183300/9, 430998386184755400706453084942550/27)
(602300360660721077500, 6867008539101162976085264171650)
(643944123233651777500, 8573645426318872538838356171650)
(160684573168093693630, 8244231467966299314661028532070)
(537592524816391739620, 4299484989696686381887481870050)
(433850517183061130050, 148532589737624087218767900550)
(451976127109096771300, 1019332908183239607975647150650)
(87304763654441914587580, 25795371655594326994545201062210050)
(2199497585098921897135/4, -38232386659749252423458018243875/8)
(8275754486899609397500/49, 2749681567790306156963357907875950/343)
(25254975871866664415260, 4011816746699707529463705943953250)
(-2403530799301581998195/16, 962094887728520625484636605166075/64)
(1776924048110510039203/4, 5642969601454184299749489927203/8)
(129053923539548578351150/289, 3939611312512020864490956207795500/4913)
(588565463465583424090780/9, 451507780687142071225219023903695350/27)
(361399592272156613211584099317308902158042770692484100/1424348406562308153354190137953449, 297398594309579399002665654102656863133462329562366794626124443594659176998416450/53755678900038047314285515018101565307351626822293)
(18464239219544991560097369042158751603292040048153297680/33971387448806191711822093996867129, 28367905274622673546705585859748071490893941434251041583923747493636715454765084490/6261378055524211651057143381008101280895045650788883)
(1956503551675276404316971121201599793363608874145314780/3383768891911755998497734985268889, -1161112108539894538828195615611202847454665838639040142738948915974567582590771910/196834417909836830647489378375908442526775433809213)
(55664881434709262431670574877823714713921366267686839020/110270370143037103760562402321181321, 111159823490030489951177671032328281006034130964747569966957264079160871030620333450/36617462759740868240460575249777941769476816167368869)
(13560296405081088889082518543068156591236525253831268150/43890580129600724386627435632502561, 34839745044919031589623930300891464450750950463511662365992578329828103664984167450/9195111118422108933214727349957944902785510661514191)
(37030392758459178914790112353104932923117815558645657260/101176626171287207105354385593825569, 60363026328139735875075702592111052263042874185186231196974340081752455044746633250/32182537949356988799769820736976139061870517051688753)
(-552466623201962561540231964327573126623645286350237485250/966144059466377549017678838420375761, 15514347624315989400281757516947653260133220889856538307068664768076836212672533159900/949648380197523963087703181995085660149087410104602009)
(192872541138606215882460549687834397883326961391973249660/2187052857228956834576359611054708841, 32842423302303246089519047018871422940318619515305388734157054517312668394970634493110/3234364182462051896657024024701189509489665239019416139)

Commentary

Specialization at T = -15/604 of the genus-one fibration with 17 independent sections behind the rank-30/31 records #273/#302 (the 'wgxli' family y^2-(Lx+B)y = x(x+D)(x+E)), with 25 independent rational points.

last edited by hdeping at 2026-09-27 08:23:21 UTC · history