curve #6130

y² + xy + y = x³ − x² − 417021599588606487x + 103355942904397210370672175

a-invariants: [1, -1, 1, -417021599588606487, 103355942904397210370672175]
c-invariants: [20017036780253111361, -89299534579322524249121758305]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 2046819555713424945668705479118862619827345878
discriminant (Δ): 26652014049742757487742217085414868043493918909079552
Faltings height: 8.9243
naive height: 133.3293
primes of bad reduction: 2, 11, 17, 5472779560731082742429693794435461550340497
regulator: 316975927372837026749.903161297866962283105628342513281352075549799550034
submitted by: Michael Rubinstein
submitted at: 2026-10-29 12:55:56 UTC
last updated: 2026-10-29 12:55:56 UTC

Witness: 17 independent points log in to add more points →

(456957130305, 308896105560821754)
(-346582105905/961, 428463274063444754/29791)
(10153687119/25, 123721074552118/125)
(762222083863/7396, 4986939883064969777/636056)
(3393401421875/7921, 1285799596596936206/704969)
(18125491757851/361, 77161023812795187658/6859)
(19816003770237/3721, 87590129193752316000/226981)
(21161344652491/19321, 83228807613506801818/2685619)
(50951863197595/167281, 146632435585166593390/68417929)
(193910262794765/52441, 2661514826306119736546/12008989)
(1407978328375753/1149184, 46610654601257464436995/1231925248)
(400689125489251/231361, 7527654781607032202142/111284641)
(3180272958899347/7789681, 23315833512561493471122/21740999671)
(3406616726597747/2217121, 183510528148814549858830/3301293169)
(8319396160217223/20693401, 77617317063987303449078/94134281149)
(15969774081473459/47128225, 315972045335046636533918/323535264625)
(1076514456328940311/2766654801, 6854938445491334852396098/145523275877799)

Commentary

Mestre ten-root construction, roots 0,2,31,38,43,68,75,83,125,137: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 12:55:56 UTC · history