curve #39

y² + xy + y = x³ − x² + 31368015812338065133318565292206590792820353345x + 302038802698566087335643188429543498624522041683874493555186062568159847

a-invariants: [1, -1, 1, 31368015812338065133318565292206590792820353345, 302038802698566087335643188429543498624522041683874493555186062568159847]
c-invariants: [-1505664758992227126399291134025916358055376960575, -260961525531561099457995721578616998276609112811677665548304369308086430625]
rank (lower bound): ≥ 19
torsion subgroup: ℤ/2ℤ
conductor (N): 565363502863493379565569563649307108299188776386222378070363275106421667133549554450
discriminant (Δ): -39412228697608401237498486986030933067178933520817917493127770273615403601711081236922902433264711434650985878886431089797471702169158680576000000
Faltings height: 26.6161
naive height: 342.7010
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 47, 61, 71, 79, 83, 101, 127, 137, 229, 241, 293, 311, 523, 541, 827, 1009, 1753, 3331, 4091, 6637, 9007, 16069, 44263, 52609, 67157
regulator: 754818738046468728836.6738481003770253068141192531193418878223715648421956502
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:17 UTC
last updated: 2026-07-01 22:31:41 UTC

(-499155234006326082923757, 402509376386904307636088023311932246)
(14139901902190764472167779, 53177370026792370019660919624681843910)
(165026138566397083648829, 558309596396209485460849837556760960)
(463343337635697262499219, 645016881525945325912024782026024790)
(-428490311544140084738931, 458176166275251046654351925125281840)
(-3076708833879670305775, 549492732729622717403028037016638716)
(-37665731401560276357421, 548455893205160937812792765224762710)
(-589425676522172250643765, 280659706984671022310703064556433246)
(-481732375371853545253165, 418490225997586738600806619185884118)
(3690858926941989311074375079, 224228684581316592254479552957274125288710)
(6130049174458433978294529, 15193625860788829982804926312716459660)
(99629044314032316202835, 553310835513576760020037678256777046)
(-336069400109872546513581, 503527960099926164391149483607861590)
(3296316187302021571885259, 6018489996489383086719697427657451870)
(239450610550684059490919, 568576446830597022058144468592457990)
(1116036406771517906484819, 1314196172527374653201729069765173590)
(1104492511976272577147219, 1297713008801587828896903435227547990)
(-556915090457934898225581, 334424588793955899397684278466715990)
(142518206539016399870550419, 1701397854372553501423223969798389683990)

Found by Elkies (2009). A curve of rank exactly 19, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:32:11 UTC · history