curve #6412

y² + xy + y = x³ − x² − 454565068774746118966929953510753x + 3730495290129351014075956669931418116003864405681

a-invariants: [1, -1, 1, -454565068774746118966929953510753, 3730495290129351014075956669931418116003864405681]
c-invariants: [21819123301187813710412637768516129, -3223147930671759177975571707475583555370468888186033]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 1150660788551932761609211252461298678005146490
discriminant (Δ): -672561463165385061573214110956584844728149154838205576635959389750038925681196095930421267660800
Faltings height: 17.4448
naive height: 237.2044
primes of bad reduction: 2, 3, 5, 11, 17, 23, 29, 37, 41, 43, 97, 163, 251, 439, 5059, 430695673, 137984182321
regulator: 588965431846153.92307235701788265879300510592999662202582730354
submitted by: Maksym Voznyy
submitted at: 2026-11-02 11:59:28 UTC
last updated: 2026-11-02 11:59:28 UTC

Witness: 12 independent points log in to add more points →

(102766819397589704748951/21025, 32943918008362314932878355110500964/3048625)
(60443533066876885129575/4977361, 387597987166535806663339884930812/11104492391)
(-129958177918318623895905/25421764, 311884713517233589225177805193295451/128176534088)
(291921529723116260522719212710175/23723352221729161, 52864561504682304774769127354090657151021875828/3653962494840471247343909)
(1258229409355986100718374543/50368376041, 31900391318058511379867821613643553842164/11304124266505589)
(1970887453253280163375/170569, -10161804230499914866389919240396/70444997)
(1323966071328966095855889523335/20648454019249, 1448105268661577736795955944982372401854804036/93827752025346553193)
(212733839697372761989240204063/15047734897881, -21021678205563614650932913194836993254049900/58372285399501000221)
(68959874707378162055899575/5417107201, -32916591445638906658214632394733065308/398704507100801)
(79012804815496795552826684655/2917602193801, -16738974338281064826977382265047483551731892/4983553389629294299)
(197154807105867415423/19321, -1055779117833044331563396362836/2685619)
(1595120506608547016375503/131308681, 51427578469002659537621272714798124/1504666175579)

Commentary

Voznyy (2026), t = 3507/601 for E = [A, 0, B, 0, 0], where A = 13*t^2 + 3711, B = -1189*(t^2-25)*(t^2-121)*(t^2-361), own construction similar to https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/3-r12-z3k3.ec

last edited by Maksym Voznyy at 2026-11-02 11:59:28 UTC · history