curve #45

y² + xy = x³ − 560715933702165990261993692150795879540x + 5299428030171662962897867758309003693598430128674403539600

a-invariants: [1, 0, 0, -560715933702165990261993692150795879540, 5299428030171662962897867758309003693598430128674403539600]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/3ℤ
conductor (N): 37782801949200843416322963769043529248461782217859568090
naive height: 279.3530
Faltings height: 21.2021
discriminant (Δ): -849674615711570771215349016224804046891153693433134502714153993581965535857133434327195120355094716597801586176000000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 113, 197, 317, 3313949, 2831657657, 4864617187
regulator: 473362182709.7477529528683850131310786537924858078223167486334695
submitted by: David Renshaw
submitted at: 2026-05-28 05:18:56 UTC
last updated: 2026-07-01 22:46:37 UTC

Witness: 13 independent points

(30978765264695144680, 132887030666136666891811891660)
(42714959134973152720, 243484714136201166251098454860)
(17517044808100174120, 29195754303760059255051062860)
(14407536544275712180, 14545360491234198754332073660)
(27024189809840210680, 99410687206608213450615379660)
(15223373261754620200, 17072594325815595384787766860)
(-19625425810319741720, 93513850432355415588023731660)
(18799275311571587070, 37447008534737807545137468570)
(-18348119874544858160, 97008001583008247645310994660)
(16004432727240519730, 20612929621400491642269765910)
(-27235206268166134070, 19202914916421997002398631160)
(13326569619771689680, 13920236659741078875971506660)
(27069452184006616180, 99781951498337473669562238160)

Commentary

Rank 13 curve due to Eroshkin (2007), listed on Dujella's high-rank elliptic curves with prescribed torsion page for torsion group Z/3Z: https://web.math.pmf.unizg.hr/~duje/tors/z3old8910111213.html. The model is y^2 + x*y = x^3 - 560715933702165990261993692150795879540*x + 5299428030171662962897867758309003693598430128674403539600, with the 13 listed integral independent points.

last edited by David Renshaw at 2026-05-28 05:19:16 UTC · history