curve #4269

y² + xy = x³ − 17919687764706770704265886823698979513364x + 350586152076425460834320390688705909752626101552870327003024

a-invariants: [1, 0, 0, -17919687764706770704265886823698979513364, 350586152076425460834320390688705909752626101552870327003024]
c-invariants: [860145012705924993804762567537551016641473, -302906435394031598162143035074100793516976095592986289055574945]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2029002416341262551558890376876260685313306986997405814 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 14
discriminant (Δ): 315176793892270922360434804847978261806955483152613374454606561871597256154463045907975556675309327733745844328114416062464 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 14
Faltings height: 22.1983 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 14
naive height: 289.6738 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 14
primes of bad reduction: 2, 3, 11, 13, 19, 41, 61, 89, 127, 199, 269, 271, 283, 293, 383, 443, 617, 773, 883, 1249, 1289, 1307, 45673
regulator: 39089511090423.7549688370287674270502587368899389660955250016064986
submitted by: wgxli
submitted at: 2026-10-16 00:59:32 UTC
last updated: 2026-10-16 00:59:32 UTC

Witness: 14 independent points log in to add more points →

(122733725362400010184, 6622116830633155018684806412)
(19467426095698817884, 95465874600355425604485240904)
(19680603477105202360, 74422410483189869248871057692)
(5723950858244824180, 498199033747055074571973012508)
(6897113247117147688, 476781011644273413695124731116)
(-3289072746247952360, 639913843840931997143409958588)
(-7504848148508857220, 696166642385599694994298944808)
(122759933899526166136, 27547027955657962231861817692)
(14333957400727393534, 310919924373324691343419320904)
(18235544865103521688, 172843398899871089073000457516)
(19405811678093074560, 100737350559660886326621134892)
(-142624938899163173270, 71568868563189721026397404862)
(-31283465470070843912, 938381776784614974305829709276)
(301559104009923748600, 4729681887984401027284987956268)

Commentary

Specialization at T = 180781/50546 of the rank-5 fibration of X312 given by: y^2 = x(x - (29T + 2)(400T + 49)(221T + 86))(x - T(8T + 53)(25T + 88)(392T + 221)).

last edited by wgxli at 2026-10-16 05:27:51 UTC · history