curve #9

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y² + xy + y = x³ − 19252966408674012828065964616418441723x + 32685500727716376257923347071452044295907443056345614006

a-invariants: [1, 0, 1, -19252966408674012828065964616418441723, 32685500727716376257923347071452044295907443056345614006]
rank (lower bound): ≥ 23
conductor (N): 113964706497998622564257410134352232583764612126645513242486361984405434455972952111271430 ★ record for rank ≥ 23
naive height: 269.1762 ★ record for rank ≥ 23
Faltings height: 20.2687 ★ record for rank ≥ 23
discriminant (Δ): -4779639209650129827198370598581502977994781114856713955972654268540889198350896060545934759063721850651300750000 ★ record for rank ≥ 23
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 199, 17858193374841750901974789649, 1006218106634655545344494448610726356220703995276273
regulator: 14314346672172296526006093.9515872795609689097642740408996297487603077991878415316140
submitted by: David Renshaw
last updated: 2026-07-01 22:29:40

Witness: 23 independent points

(16902136044621724275584661392595/119224493521, -69455519784971993679807552308609739430858248812/41166906143372569)
(6647882272466103821634772046571/30891226081, -17137023844710987140049387309945953892946213544/5429411004770479)
(1277229332035649706664846727592/2961427561, 1443380843339272397458721030742392016696304046/161157926442059)
(1754834771916476982132090651/369369961, 49412130720987886904443301152758710388796/7098921280459)
(902743031953703698667092998/307406089, 6538434104009303265024749952830709029353/5389750958437)
(103579510135061476534950819/45091225, 230697883363551870088729854504374414548/302787575875)
(31762044569407766003397375255/14054813809, 1411381089291349753164768808558921002947204/1666240341498377)
(29436984213667648723395/17956, 5657335012046240705357319452802233/2406104)
(1127027270330215920, 3523978127407100674110377602)
(686464244502821899711515/139129, -394563651945882403580468873435105816/51895117)
(11962675953816366561795/1369, -1167962768316319592876571517317044/50653)
(30520680805402695175757355/3345241, -151915114589061403100759698106532333112/6118445789)
(11449775538050756019357635316/967521025, -1150775031908416918955115365651634494501651/30094741482625)
(4969418243982621661795591770/1285294201, 184569435055535326363669745422918052707327/46079082400051)
(480465113537612829840777315/160801, -10531550647702714814852169224678207441368/64481201)
(97907154284679777917982542166035/57601436172721, 964874722537391293613786748114488474882993683572/437169613520472565481)
(249989354826313432718977195/4397409, 3941156276776007263792745630379334937996/9221366673)
(54840074123086507808388135/3996001, 387496978790653709721061294119215460988/7988005999)
(9690141319063801580189469420/87590881, -953144078079942906360903670036536669593542/819763055279)
(882142442406602738753880/76729, -775394556680837651292166377698874734/21253933)
(2812175950395226936581984/24025, 4712624271973109965160039085789391367/3723875)
(7126269737101017406079752337071371/2947180538019481, 83015454575998684006900205726968222686505350799684/159996363164349841378621)
(2143448685801212487450/841, 10099849221189668277354753748208/24389)

Commentary

Found by Martin - McMillen (1998). A historical rank ≥ 23 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:41 · history