curve #5997

y² + xy = x³ − 1245707854119705946806595120095x + 535070766333003318989540184240454514437640025

a-invariants: [1, 0, 0, -1245707854119705946806595120095, 535070766333003318989540184240454514437640025]
c-invariants: [59793976997745885446716565764561, -462301142111714957297928215802580870548969628441]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 161629801809854228923433025145794886883310
discriminant (Δ): 34859386028472206375655464503865538368668059475459230124791779354656748474411478528000000
Faltings height: 15.9986
naive height: 219.5054
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 31, 37, 71, 109, 149, 331, 853, 240043, 483139, 594630419
regulator: 2011275737849.40250258929017207206351769323663651105534133778
submitted by: Mahesh Panchpor
submitted at: 2026-10-27 17:50:30 UTC
last updated: 2026-10-27 17:50:30 UTC

Witness: 11 independent points log in to add more points →

(11353498438244625958354830/17442749041, 190986835850760761809126115648493915/2303681308593911)
(695607607404390, 2265064790810337915405)
(8187856952948467561955190/12584576761, -12256153373586471323697585004214295/1411750405625741)
(11374534100640678830010/17447329, 13664012034027916868331114945765/72877493233)
(242341403113543654810101510/346218090409, 501818303959900542444062683633304791935/203715763050926827)
(2314942798943070, 100284666551866758314205)
(26927489064117714060/38809, 16705395685861380392377767465/7645373)
(1963234032524527071406770/2966871961, 115454902126046634092867808140291025/161602548843709)
(625909045086447068815634065461537510/956408580033082205929, 10281188113696952302362213743464135222453428103793985/29577754188726259882445821849717)
(47100569385025848150/72361, -1643901524928780720224342895/19465109)
(5927806054558507914377190/9087618241, -185703911300301910738931693407276755/866313559296289)

Commentary

t = 709/1079 Family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3 (Z/3Z K3 family, parameter t=709/1079). Family construction from the Layr Labs Elliptic Rank Challenge public repository, submission/board/3-r12-z3k3.ec (commit c8f82607a219df846ab80dd9e14ee09a2904d709, 2026-10-17). Not itself a repository curve: this t-value is not in that file.

last edited by The Wren at 2026-11-01 22:49:28 UTC · history