curve #74

y² + xy + y = x³ + x² − 215843772422443922015169952702159835x − 19474361277787151947255961435459054151501792241320535

a-invariants: [1, 1, 1, -215843772422443922015169952702159835, -19474361277787151947255961435459054151501792241320535]
rank (lower bound): ≥ 21
conductor (N): 26112076915897777815571388664430310998157918697219343275140810790098571234096793308930
naive height: 255.6932
Faltings height: 19.3584
discriminant (Δ): 479737754043767746536923774462246533556793859277365678757052823009411494064048885744243744500948985175040000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 47, 4507, 115482611374267602141168398241396608699381902319617225736297616061235976719
regulator: 1057662683061657998079887.5644599317446350524828508690354387929045294397158378546
submitted by: David Renshaw
last updated: 2026-07-01 22:29:41

Witness: 21 independent points

(800843008889340065933/16, 22662214190910903990783584765347/64)
(10610541066763914590637/2209, 1087744114825178454840094794778034/103823)
(907186946780634143, 728916386168451830641677698)
(196833201085564442194083107/227919409, 2277807398930440819587410184793923763894/3440899317673)
(185463474139064652528000075/366301321, 225699857838583242849473830466481978146/7010640982619)
(-12485261071234691432503/123904, 1543303353428939982282171752702539/43614208)
(-59703014087684747037/361, 741881245094154068525036126962/6859)
(-73270463404799613067/361, 866878137858638792891117943482/6859)
(-360733396398627565, 106985840484096728947883974)
(-389445180957906897, 74288355118790673852542098)
(-1474458350349858512665407/14205361, 2278493401578368084310409028259332632/53540005609)
(-114305856035468892691779277/278589481, 16972779768877136292841029639987095378/4649937027371)
(-21972533600828202797/81, 100790786584963504563876005302/729)
(-25047938415396324842058977/71216721, 68347192566984943007522052612937752062/600997908519)
(3434828081885118352213715284707/5137262501809, 4279912483838925044234939165329697576812433846/11643877735262694377)
(-227656313261676647, 133660024327268949095297798)
(-4098089434105992137835293/12552849, 5660088413991351759301403659890889706/44474744007)
(2657828735869178020212617/1495729, 4174499731549997186596131721273201376/1829276567)
(883965004314243424124994323/850947241, 23250077986002214917145041708721276812178/24822981967211)
(37543938954172817209003/73441, 1224097915991280099903835490020298/19902511)
(19165312347502458410162233/17214201, 75593839815741485450348997055551694952/71421719949)

Commentary

Provenance: Nagao-Kouya (1994), “An example of elliptic curve over Q with rank >= 21,” as reproduced in Nagao, “Construction of high-rank elliptic curves.” The paper gives this minimal model y^2 + xy + y = x^3 + x^2 - 215843772422443922015169952702159835*x - 19474361277787151947255961435459054151501792241320535 and 21 independent points. Three point coordinates here use the values from Nagao’s scanned paper rather than the later HTML mirror, whose P2, P12, and P20 appear to contain transcription errors.

last edited by David Renshaw at 2026-06-01 14:27:16 · history