{
  "id": 6168,
  "curve_key": "2478987552816332282809:-98807008307038966988037560386477",
  "ainvs": [
    "1",
    "0",
    "1",
    "-51645574017006922559",
    "114359963314028339882885488382"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 147.78641155459536,
  "faltings_height": 10.324032935328939,
  "conductor": "47290188619362468646572850422665780847445759150230",
  "bad_primes": [
    "2",
    "3",
    "5",
    "43",
    "293",
    "673",
    "724802984897",
    "256494866860501002324469233539"
  ],
  "discriminant": "3166373766936588196103418564163447860905381658541344486365700",
  "regulator": "146091460021831548521769.66548677958100218584635743082784575155171672377621",
  "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"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "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.",
  "created_at": "2026-10-30 14:39:17",
  "updated_at": "2026-10-30 14:39:17"
}