curve #6242

y² + xy = x³ − 24136902500908920034530x + 1410815937040627157553857379662916

a-invariants: [1, 0, 0, -24136902500908920034530, 1410815937040627157553857379662916]
c-invariants: [1158571320043628161657441, -1218944969604839721106598218271245585]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 165833851234486775281739639418068527351978197028047383177658
discriminant (Δ): 40109355342009178495432474492917876619702110438642360030238831583232
Faltings height: 11.7539
naive height: 166.2277
primes of bad reduction: 2, 3, 1533109, 3217741, 95524043, 64377117305383, 911073883732752768320363
regulator: 1666174659472690105.212198130148132984965205055091465990867481679285630243545
submitted by: The Wren
submitted at: 2026-10-31 20:38:20 UTC
last updated: 2026-10-31 20:38:20 UTC

Witness: 19 independent points log in to add more points →

(3099001434872897801418522/746497728001, 5451707054137311802716965411290415394/644974783490592001)
(1313790581472, -1495782243243778386)
(335453497980, -176245084751722194)
(277646858466, -126935004706920894)
(41101721646, -22094882964608574)
(-13201439778, -41559072806204286)
(-31884975882, -46346554336533726)
(-153422073930, 38764103162250786)
(-157963890540, 35804640888986766)
(3604114549350564/58081, 172564281213416845007406/13997521)
(-19304816946972/121, 46119554630933821194/1331)
(-106806010264945356/779689, 31898277491622003292525386/688465387)
(139833643532654166/546121, 44242353597575046541410582/403583419)
(-83415444296664/625, 741959325039508724634/15625)
(659027954383812/4489, 9659480635710026828922/300763)
(120942684708, 16145492500321998)
(173222787176258364/1723969, -132704905130230552012434/2263571297)
(26170244723025/256, 14001551263212858609/4096)
(17593963476, 31489645445621646)

Commentary

Original search by 0xArion (2026), public Yukon elliptic-rank challenge submission 598e1604-cd20-4fe0-ac6a-98105a476e90, validated 2026-10-31. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/ab6a5c670a3d01d0ddcfb34c37f09b98852d387a/submission/board/own-ec-r19-b3178d0fc424.ec . The nineteen supplied points were checked exactly on the displayed global minimal model and certified independent modulo torsion by an exact rank-19 F_3 reduction matrix at 21 good primes. They are not 2-saturated (the F_2 matrix has rank 18), so the certificate uses F_3. The rational torsion subgroup is trivial (the gcd of #E(F_p) over good primes p >= 5 is 1). The complete bad-prime list was verified from the minimal discriminant, with every large factor proved prime.

last edited by The Wren at 2026-10-31 20:38:20 UTC · history