curve #4020

y² + xy = x³ − 160256386832398408740279442805457976320x + 5222473749681836423342415380162974219673618277066709538400

a-invariants: [1, 0, 0, -160256386832398408740279442805457976320, 5222473749681836423342415380162974219673618277066709538400]
c-invariants: [7692306567955123619533413254661982863361, -4512217319725106669779385348312742411227306311267630015472641]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 120600636120040262281305547153244997893031818350820655996701181065730
discriminant (Δ): -11519062039569966025131968526535847841171149512205661966141121039937344909427885169699139628879923280398476614970300000
Faltings height: 21.3345
naive height: 279.3238
primes of bad reduction: 2, 3, 5, 13, 37, 61, 127, 173, 313, 337, 491, 541, 1721, 2161, 2423, 3469, 3907, 4547, 8221, 13477, 203713, 1365868604929
regulator: 9356738396160567941259099197.61023494219477993934445511574161376253482881
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:47:16 UTC
last updated: 2026-10-05 04:47:16 UTC

Witness: 17 independent points log in to add more points →

(155809128845053332651432123822990/194720095441, 1944631003872511576052032806000857169062781889530/85924331235345511)
(4766554437646678525980094934662848251636340/105240722081741309011081, 10291905599022211887030131810925384086554466863144090434184006020/34140960006211691692978670260024229)
(-1199729188402727897586869482952233273020/60708618868137163561, 12258174506899734610127297894861204894159218964021666234040/473015670911975646286163481941)
(374708261092013727138999920121301470/7237486620349561, 226816702810136272709307526606095706838183123216433570/615717731759892129419059)
(128417915032386701694199580877652826476353803215/888265146492484904756325636, 45882272504822624198693747214815394570262433692573350767141814212866805/26473656686076507958149642211524303140616)
(91210087328886798322244286450674838448595/962235239395639245124, 27384690760884845363933323165041023963638154581303614617094535/29848457085401045975682792095432)
(2619698196768221762067561439148520470406154403940/38488245392633695637976890929, 4201709939493912068715246869568775474783047175950181400217114296128484760/7550786128397260362081803259373303904482217)
(1845789449772326951965482993124512088870636740/268107701388052658106696121, 292700649115562229044678769192515432973767778338690856983855785123980/4389994072980659607457537839589456485331)
(23846837557710128235506262834212133719996929041083551140/1057503904864656585285075183721892881, 124352214187757262616033524307638964414724331904290168134658247785042361622870719080/1087484234334929026727076825571792273748266286710286471)
(-1078545282732900792165384645648625099652678940/73229962708693154541402481, 41510960354576760423348315884841070419475221999895262135152805165160/626661796608456569505904241570826826871)
(159153707663222121172618775367306737896840378350/3450916452698405462799049, 63492894248204029125681631211308652218147617731837827800307461003841740/6410644111812279860633190303261985157)
(2952486238547264185972740690972634485803084137547830635/8982468720011874377752730600170084, 5069807471520129376111557852146282550225204346163472697685797176213691626629767635/851321438653301612939033920771547549208531437912552)
(70093507554512306215840743039432058604197608775039305806810/1651857284926731729667363756083522077209, 18364780690876419216486135764410103632139178466817583007730090369560408917163830996732610/67136513583198000947192857881067294196574278616060052983677)
(2430041657321699589170819328192326829381224021881541712340/121204479060116038488430986407117896689, 133894110741425263287521277212368492384193140904830836895267987506398631460405004894920/1334375329489090350008990888354751923524017023553266056887)
(62363811452625630613330485822857312408334079889006235323729607820/5026727184799581149828124067819112610021412449, 25560688279149251497041660773661648298120945149042410653987987076765837523610953531856437734397080/356392021639710364939178406311769630265268097914177905050303959508207)
(344218794655401370691487929615618413442622080669425092512275352096700/46973691653422868769836879953755964482549853390201, 21456209642875556433738922729777760408166495209060209404074418237773241820971406487126158627395102962520/321945262675548039930718216579289621493832709764229527902654749017152944051)
(3380268153953864261142452351652156869480293479700872383611896612410206210186947200566273660698611583745816412607387351386781009984198976265953703626801/156256918603237565694722476696668313192212222734131540092668196418331276663362820862091210408510960129911641089601353858191264998400, -6732158923609081120793355789644744431742464086043508449168888333180955083383763791379240891234216218874152271102801115334598193020365983554032384794719526887211137271809117689775504867171043410417388671911501407105966072135641/61767337822659837256469397872771466472877132827641439837864817996545398466107640428441718249426041418848450447699119038434831373694212426812614622646815168487609592632872227816578247062720868352000)

Commentary

Unlisted rational-isogeny neighbor of curve#3693 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 17: 17 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:47:16 UTC · history