curve #63

y² + xy = x³ − 1718612993735110076283239582307203184558215x + 445677626128337788660554947611110167094039534522619288819463817

a-invariants: [1, 0, 0, -1718612993735110076283239582307203184558215, 445677626128337788660554947611110167094039534522619288819463817]
c-invariants: [82493423699285283661595499950745752858794321, -385065468974883849402843214871548112294742551077469184169304929369]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 113867782999071766513257245176182365522858404266111886760688305190 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
discriminant (Δ): 239065937769943972318715082774604147597436659174149372442869290187913170319391241801587631661352787169448770987206379287600000000
Faltings height: 23.3302
naive height: 303.3639
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 71, 127, 151, 179, 241, 281, 449, 503, 1223, 1361, 1399, 2017, 2083, 2293, 2843, 5651, 74923, 436549
regulator: 36389786154315993359.606350911054022215893750090730455834360025330154869588
submitted by: David Renshaw
submitted at: 2026-05-29 04:07:17 UTC
last updated: 2026-07-01 22:33:41 UTC

Witness: 18 independent points log in to add more points →

(-916413066082020853346, -35369782405684521707886952377707)
(195434116396250243974, 10828966775531818886070739823413)
(1265034698502036374374, 17205341075829552264529317189013)
(2216840056561938704974, 86776618821766613189523105183613)
(1317460353165617357158, 21637731566353513111501700533141)
(-419919197918370459176, -33065253185351621643390087304037)
(-29666703514651498703639/64, -17307761259547457289873127491657319/512)
(1028534464393349854573171/4, 1043092047443803329899021606077145579/8)
(-4727498668121314310834/9, -936668010323115521156024630441249/27)
(2529716865202204892214, 110846353150164494211027781928133)
(1199344199314248064924, 10470797020157142702308988437263)
(5710166898431918122165966/9, 13644960301239082757781340351692685151/27)
(-1338652300506182722676, -18640047007082369619825194657537)
(5372757156747175085983246/1369, 11786893726772068452866617397651512729/50653)
(29128426914241943990086/25, 624790324363217559708520261779569/125)
(4983719710548842292787296/1849, 9864984392366637148101983568300087321/79507)
(10589515239193964535623854/961, 34220949959199862455549906395740174723/29791)
(1023941243985123630141565744/259081, 31035258051713148261108384857840848626527/131872229)

Commentary

Provenance: Elkies, 2009, as listed on Andrej Dujella’s high-rank elliptic-curve tables for torsion group Z/2Z. Dujella’s rank-18 page lists this curve and the 18 independent points submitted here.

last edited by David Renshaw at 2026-05-29 04:08:08 UTC · history