curve #12

y² + xy = x³ − 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497

a-invariants: [1, 0, 0, -27006183241630922218434652145297453784768054621836357954737385, 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497]
rank (lower bound): ≥ 29
conductor (N): 188691278153039213628440510802703336227022485808859170041743892297765880547867276243424847942033952274950770795963060482541371967126966027206485174370 ★ record for rank ≥ 29
naive height: 436.0125 ★ record for rank ≥ 29
Faltings height: 34.2354 ★ record for rank ≥ 29
discriminant (Δ): -58514056884179895803252795545068205623215359270190478217393508483140281138663769634024489915971980794469429246265449489823479442097938729025702174957426421151737866434992604323840000000 ★ record for rank ≥ 29
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 31, 41, 43, 61, 233, 241, 4139, 678146849364709860535420504397393, 159788990966780131363155786084695062643236502969, 4402149008473369392540402625019227412319473055901
regulator: 1433744182671713097629179252379019849.49384267391518181639888988960560090198905327978618140977503
submitted by: David Renshaw
last updated: 2026-07-02 01:49:24

Witness: 29 independent points

(2891195474228537189458255536634, 1159930748096124706459835910727318679593425283)
(3402542165322127811451484642234, 1661508223164691055862657623730465560755290883)
(4298760026558467240422107564794, 4313142249890236204790986787384907722927474563)
(3728756667770947009884455714554, 2530180219584734091116528693531660545660397443)
(5991744132052078230511185130234, 10418901628842034362301273055728300669218858883)
(3236493534632768520540227223034, 1324626796262167243658687198416201825373745283)
(78226686134991174232380689386234, 690394210062759896503429654125516779999512554883)
(11492605643548859374635605140234, 35536316911450952155461624238308456029618940883)
(-5143303362384229804906088118566, 7622356511107986864120352355674305680222368483)
(443985655575065435281568435002, 6584468124388858623214803939643557365635620355)
(-979565018904269680752629749766, 8987348422104537684966706438714038633832170883)
(5184894285212178249566461261834, 7390536788003150201273204464695859875505480483)
(-4469171023687146502067179612166, 9310658892841458934133221137392081403414455683)
(3606405835110925482450522970234, 2183644666981703632482662193390480040127898883)
(16151744576785317732688993162234, 61908882092472338946519909276455831463747210883)
(3573684355943766387962362869754, 2094467155115749424853047283659077805560259203)
(-759376049938858166436491644166, 8679171135458197195914024161800061810952119683)
(-5328058719935886182106003119366, 6920588147379497633202935557367499676224350083)
(5380268474895377355583039694554, 8105660240030025092450118297303424395856037443)
(17069233487425098088940203248484, 67583677272795299213867443505411893525786510633)
(5215432542403430758248050783794, 7501515746204716855921710958364078294243814643)
(2838942178046024039763692432122, 1212346280964590308944175800544505700108208003)
(243146882395382015946366404808154/81, 811625272160726332199288136187427505366582108107/729)
(2558229016839511149831260080762, 1706598395830079994387505244133382709649637123)
(2361253942905600810977556672634, 2157503396243552448798851089310708763298766083)
(2678312077644931683114439906234, 1462722361020796436741527433473386115047618883)
(3379397084927230910084852603902, 1608494167359575995485655188349208450365853755)
(3632407730870998917912491355514, 2255654937037700801978158381185619053396712963)
(2428778263277521959543043930234, 1998325023610603606161737305486867803334410883)

Commentary

Found by Elkies - Klagsbrun (2024). A historical rank ≥ 29 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:44 · history