curve #725

y² + xy = x³ − 850332964620368789983090439976954335505180865940x + 268290998160355444898711817350536305957921672851351964981116994803528592

a-invariants: [1, 0, 0, -850332964620368789983090439976954335505180865940, 268290998160355444898711817350536305957921672851351964981116994803528592]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 20852911486978615213515524023207651425652574894610641921693383601594492947096474557986507014191220888983630
naive height: 342.6995
Faltings height: 26.5374
discriminant (Δ): 8254821159983526799865867935685823116190433307423155713720753121932515043129006286473325913283104656292864291794591134315382720609618145280000000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 31, 41, 53, 71, 140594917, 574175953721, 765239683751203, 4767301025119311657119401271590390184312602920466787397559
regulator: 169043853189814661753236103235.21740051324521313608195351196172411646389925051549153762438847
submitted by: FNacc
submitted at: 2026-09-13 11:14:37 UTC
last updated: 2026-09-13 11:14:37 UTC

Witness: 27 independent points log in to add more points →

(-9598328911134337278296, -525786923919470333461145197565099732)
(-622071816502874552449976, 746012036432604561317198597060332588)
(671660442080935696710124, -12674654326229165485118023562475812)
(17056574285999734016082613576/49, 2227594785457249396335329404263475685706484/343)
(-377595446836350896193146, 731803203662647634209334789905588168)
(10464888344453219550168904, -33725661775606434743699988555157568372)
(193334001784856150482279624, 2688174324024120021575773084727884946188)
(69860418699369952122346024, 583860520375021888445891938829389785388)
(380029929622936860667624, -4897753922994617854639193831961812)
(339878220456935516834093224, 6265899236195513857317004222770628242988)
(-696259441301986679951576, 723057842717003230684087907465346988)
(-931147634892887955567416, -502730626335615466738511858455607252)
(17251237086053143234689544, 71551775176283331446700204247424904748)
(199504104706391473003624, 326476218646132963905178352388710188)
(283789599831389841705124, 223228085165058754677232281597644188)
(-181545418940699317155176, 645508748474360052625006714042569388)
(2042841879371727664986408, -2656386446280833173822436831915973204)
(5256937682499427442137431784/6241, -190743586065456233905941793789775257497068/493039)
(1865212921914081744194237224/2401, -32547351038452785702384249415308071587988/117649)
(835008371553706719894253666/2209, -3069498968322201075002503604858695034776/103823)
(6216009841560769588982824, 15334980906842717365937115781656777388)
(1440596950177938997459624, 1425834131306998548819252013467014188)
(81728964538573179434351755486/80089, 15426365942024634414001160130412325740215256/22665187)
(825088275909184987395304, -358311500084513689493505891407570132)
(3044750023042295208156124, 5089754709466480518627928111710356188)
(68805048723959329759092724, 570678659007760418779473991485630418288)
(2112691909352169322192624, 2811001647365635055257876303054694188)

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