curve #6168

y² + xy + y = x³ − 51645574017006922559x + 114359963314028339882885488382

a-invariants: [1, 0, 1, -51645574017006922559, 114359963314028339882885488382]
c-invariants: [2478987552816332282809, -98807008307038966988037560386477]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 47290188619362468646572850422665780847445759150230
discriminant (Δ): 3166373766936588196103418564163447860905381658541344486365700
Faltings height: 10.3240
naive height: 147.7864
primes of bad reduction: 2, 3, 5, 43, 293, 673, 724802984897, 256494866860501002324469233539
regulator: 146091460021831548521769.66548677958100218584635743082784575155171672377621
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:17 UTC
last updated: 2026-10-30 14:39:17 UTC

Witness: 18 independent points log in to add more points →

(-936604331034784206/142157929, 696035081583703102770341143/1694948987467)
(-11253799938229584/1907161, 1217396767609820261357809/2633789341)
(-27074328646794/22801, 1436214749820337519324/3442951)
(113971918642041/4096, 1178790254212918561171/262144)
(951503818843299/850084, 188685085857203838203107/783777448)
(827021680565646/491401, 61820779256543181403244/344472101)
(47635518853085691/7672900, 3861719539038668364192151/21253933000)
(109678395168117129/43454464, 2661813607513538336497477/286451826688)
(41138389388081508/19175641, 9733313746470154355182418/83970131939)
(126807039873947028/20784481, 15385784792796188713402346/94756448879)
(843598611288821964/26988025, 755556655890498662918926298/140202789875)
(4376934740871353964/63792169, 9108284961438894961183227082/509508053803)
(29876943494357715171/1314062500, 155769120748014550911461559031/47634765625000)
(10797828018392783145/123054649, 35365472599943946302821598006/1365045221357)
(102769467193161788484/5886418729, 961673769387293090767070923303/451623704145067)
(1061680309662026902296/463326623761, 28294588798314321945190885213394/315377629588261241)
(4809213373252238348754/2064195966361, 242410720897321901246540308031080/2965694334945805891)
(7693131091744230716946/54200207202241, 130544549363335941842757747634770811/399026207239267219039)

Commentary

Mestre ten-root construction, roots 0,3,17,22,24,33,51,66,84,120: 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 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:39:17 UTC · history