curve #313

y² + xy + y = x³ − x² − 220168815383488457944879718815779326933313515572495053707x + 1257194842755882887406594337014748119778734089385499851686385036711122100696018139931

a-invariants

[1, -1, 1, -220168815383488457944879718815779326933313515572495053707, 1257194842755882887406594337014748119778734089385499851686385036711122100696018139931]

c-invariants

[10568103138407445981354226503157407692799048747479762577921, -1086216344141082814719297507133185911365992746312977852592828337100813775637700741299969]

rank (lower bound)

≥ 20

torsion subgroup

ℤ/2ℤ

conductor (N)

14143803817470093613466453371895612334117548667434143819728526037275098905399795800409355189270

discriminant (Δ)

249178700214336462700040090529345761697029666451736506636521278574102532187649713021760375104270067437065715167209867291412637269415937398182192231222662235033600000000

Faltings height

31.1161

naive height

400.8156

primes of bad reduction

2, 3, 5, 7, 13, 17, 19, 23, 31, 79, 131, 137, 227, 229, 337, 443, 593, 751, 1093, 1297, 2113, 8231, 8629, 19937, 33073, 47419, 157559, 479081, 858259, 2391953, 11332579

regulator

961951090651248769167861772.741981575340867610115628971712379176091047951404205

submitted by

7fff-zip

submitted at

2026-08-24 00:24:06 UTC

later contributions

rank ≥ 15 → ≥ 20 · David Renshaw · 2026-09-02 19:09:08 UTC

Witness: 20 independent points log in to add more points →

(8745147724266533162314154991, 24372398301238899242666279172749746772854)
(9296759061407832022961892091, 117706329405900284337662341402193827728954)
(9348862375787350187490292075, 126368262383594925310138413633972678942922)
(12952802258768037562920419691, 760624237746275145822530527782809048420554)
(43484757540498761633576720400171, 286751404723944208008393246650638768341137366794)
(-78861499369273690389344767875494832741/8966764249, 1346087245271730373547334155459988570340630161867317631778/849089807030557)
(-20545657343963607163229959678188398301/1378859689, 56774524050756182292718722707504440397385138559484055698/51201196831637)
(222624245683601369567481529011/25, 6553654165045069004251676731058379873366626/125)
(71404964934510555253594898735956611/8637721, 1151822561862373963670793866411893672149331554476126/25386262019)
(323035036417378141977421677278765451/90649441, 621090817915230285758693039801703470201488950760046954/863073327761)
(455729176922973426074935943292618051/61136761, 83076208232587081182675469934207566926758713391898286/478028334259)
(1711629659373439095896730981904945479/206295769, 119326282435980261523253885007180401807944509340238738/2963026130147)
(3710855353554148067793148086649547139/51529, 7148435947116837028888534359547088583159976229571351182/11697083)
(7652493874625081488190244804490033179/174424849, 20088480288591250751859796945557264191185972959490032262/2303628980743)
(85359105495888223635686233687474402419/38809, 788632704037330819609676996384868307382768715963643412642/7645373)
(91494716017880513451877751324445599451/10807889521, 6446662539756044783987771788734852708985254694783123994/1123599002492681)
(1271114230830608693055313934372378255691/37885118881, 41478458356618149674173069632364138477133262650999581495594/7373997424116721)
(71407277791357977524425503035727078146131/8429028951841, 39453092818398915161562771336778097203020218508647744091806/24471822746271986639)
(724419259817513326479501997201115825998321/86875280717481, 26814902645448999631723382637392521381612080397524117812189984/809737647105898699371)
(2666067840675737622507261624932580267369924051/24709579830081721, 136421282650199878011169477810930917405611016451516828103599072998014/3884168312119448083707731)

Commentary

2-isogeneous to curve#40 [1, -1, 1, -244537673336319601463803487168961769270757573821859853707, 961710182053183034546222979258806817743270682028964434238957830989898438151121499931] found by Elkies - Klagsbrun (2020)

last edited by David Renshaw at 2026-09-25 18:22:29 UTC · history