curve #4121

y² + xy + y = x³ − x² − 300611306490897883327602884168021573431887732997919030x + 50193876987428593579965421341408693217589976084815330130570661564868063188950597

a-invariants: [1, -1, 1, -300611306490897883327602884168021573431887732997919030, 50193876987428593579965421341408693217589976084815330130570661564868063188950597]
c-invariants: [14429342711563098399724938440065035524730611183900113425, -43367509717138304853090123974045068737963796538518222252520391730758256267702805625]
rank (lower bound): ≥ 21
torsion subgroup: ℤ/2ℤ
conductor (N): 4178638861833190752303116809257681047270030756539116341979925537516195581325565936225850 ☆ record for ℤ/2ℤ torsion, rank ≥ 21
discriminant (Δ): 650193191654509113292632972797762680738505880457140731283181487678333381687750234005100690109996304140030238294982345098548639351076392973234656821882338284000000
Faltings height: 29.7630
naive height: 381.0266
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 29, 41, 43, 131, 151, 167, 179, 197, 251, 331, 389, 463, 607, 797, 977, 1051, 1427, 4969, 6971, 19381, 125921, 1644949, 6797369, 178110811
regulator: 777214800652131011390211.59333589417485663050547035886556923594270213227754152211
submitted by: wgxli
submitted at: 2026-10-09 20:20:43 UTC
last updated: 2026-10-09 20:20:43 UTC

Witness: 21 independent points log in to add more points →

(186231711179911580010839669, 818201558933054139955900086814002275715)
(188440772098619769768658059, 487819533687568935377863258203862196145)
(52978747169735012679145373, 5866563266401428982190141762049124902255)
(93750995821405850582016929, 4778626216329987729449783295361770935835)
(-73110213707414116106711257, 8472358125088793377727246275651090093371)
(164455682879852171634121169, 2281327178236850821452997960119263249715)
(446904910486247779238558193, 2259832708737987480603212748636948700371)
(504111068415405786529937309, 5173114362940898820570994477403960689395)
(574880584743429709896987809, 8207875968887605872598326206644209091395)
(-512588717467991033803657465, 8342816923472956417793425917438946065021)
(-559712657248815031449995941, 6565365451815383371085695008734675164545)
(653346090275605835410596273, 11518623828888491921034826680300367870355)
(871914415575186787900135529, 21235493435567867945768855345343472247835)
(-330266798739357909976168341, 10651365039756448388950148736272685115395)
(1733878068251294199373548059, 68859172590741223250363811261472618687395)
(4534987489506891468884893809, 303239152912559209426474461158368701075795)
(3880873107991205409010690861/9, 23333846401670953069910483147345256415345/27)
(18278183934251475254032348091/4, 2453954669522252860246390801327077234400305/8)
(25054093391287632812323916915/361, 37358044591645590904803847257672044816493057/6859)
(-131543401836019080587796565519/289, 47308372899137041363981661362041553908300555/4913)
(1275359983221308397077669347211/38809, 48564991758714766646673286661550684802267566855/7645373)

Commentary

2-isogenous to curve #4120

last edited by wgxli at 2026-10-09 20:20:43 UTC · history