curve #6113

y² = x³ − x² − 674131219219596635685645014991889x + 6690306155949934356049374006657399913576092265969

a-invariants: [0, -1, 0, -674131219219596635685645014991889, 6690306155949934356049374006657399913576092265969]
c-invariants: [32358298522540638512910960719610688, -5780424518740743089476868006508162447863979400133120]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4839954604749099000664203083594381012438592
discriminant (Δ): 270691963989245919978429504052914521480886396206053429237856658080489271792280692477131799566106624
Faltings height: 17.7150
naive height: 238.3865
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 73, 257, 431, 1741, 7333, 9067, 16381, 154571
regulator: 1509389547620.0282644854206329413212052896724466657671155167623
submitted by: The Wren
submitted at: 2026-10-29 02:46:15 UTC
last updated: 2026-10-29 02:46:15 UTC

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

(-20748726704338541, 3427118806018673508164604)
(-19129718467060928, 3547647132212205772654905)
(11899297211284405, 594540890900446294849968)
(43620668554081893, 7764279351574274401032160)
(-21694207514283076499/729, 12410365864046470344047742992/19683)
(-602648564109378547/121, 4193033113001488374257299920/1331)
(-6863841667612275/4, 22402908331055395362233329/8)
(146342796461497099/9, 4510850607405331847517860/27)
(1307920290533817616/81, 89472677579257461788414665/729)
(1598054712882320536/25, 1874113864646113362349686609/125)
(3676823001542245381/225, 656976047343129283945704704/3375)
(3921490511007834831220873618252525/422778782872934569, 305545469095755800904092322455297547862346948844864/274896858134579604369742997)

Commentary

Source parameter/search by bufferhe4d (2026), public Yukon submission 15009775-297e-4a19-9179-4a455b778310: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/6837b3bcd53a73fb5af3e271d91dec3148236631/submission/search/x400a-opus/xlog.txt . The source log records rank12 at T=1364/12167 in the rank-five X400a family posted by wgxli with curve#4252: y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q). The published section code credits 0xArion submission7e9f6673-cf5f-4cb9-82b4-a22add86d720. Using its four selected sections, a local seeded PARI/GP replay recovered the twelve displayed points and bounds[12,12]. Every coordinate was checked exactly on the minimal model; independence modulo torsion was separately certified by an exact rank-12 F_5 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 02:46:15 UTC · history