{
  "id": 4162,
  "curve_key": "686968427913619804729:-19154068641253988812159434388333",
  "ainvs": [
    "1",
    "0",
    "1",
    "-14311842248200412599",
    "22169060926184648345241903566"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "naive_height": 144.06013588913103,
  "faltings_height": 9.952571088673112,
  "conductor": "781869987821636658363933019146211848405454531611164070",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "35266397",
    "236451511265701",
    "240417872516149973451262990529"
  ],
  "discriminant": "-24699272915285502037716644074828832291128308653596672971300",
  "regulator": "34366661978483124.5014355097619062424133222199111811632176531104906832341",
  "points": [
    [
      "-4204749791",
      "-89480484328828"
    ],
    [
      "2180406313",
      "-36462792086062"
    ],
    [
      "-1382079529096523/388129",
      "40447882224779851882066/241804367"
    ],
    [
      "6536163778447/3969",
      "-13846435370658239894/250047"
    ],
    [
      "56945596196656/13225",
      "-305614529470961588866/1520875"
    ],
    [
      "669051101617/529",
      "-949600754867828324/12167"
    ],
    [
      "207736156259248/57121",
      "-1842808774377294934853/13651919"
    ],
    [
      "-104266622324/25",
      "-12062436965478191/125"
    ],
    [
      "898706472230599/292681",
      "-13412095972847957306578/158340421"
    ],
    [
      "-112954054776431/47961",
      "2173296425611338914438/10503459"
    ],
    [
      "5377110968206846/9409",
      "-394288668415162624555579/912673"
    ],
    [
      "7555671349",
      "587680602958958"
    ],
    [
      "3463880271",
      "118976435913526"
    ],
    [
      "1650720169",
      "55155532570628"
    ],
    [
      "9016083123887977/82369",
      "-855601730234361334536836/23639903"
    ],
    [
      "203265823",
      "138810383799278"
    ],
    [
      "911076797701/361",
      "316112074424219444/6859"
    ]
  ],
  "submitter": "Andrii Prokhorenkov",
  "history": [],
  "commentary": "Mestre construction: sextuple [-345, -264, 33, 69, 195, 312], T = 271; y^2 = r(x) with p(x-T)p(x+T) = g(x)^2 - r(x), p(x)=prod(x-a_i) on the locus sum a_i = 0, 12 p5 = 5 p2 p3. Found by Nagao-sum sieve + ratpoints search on the quartic.",
  "created_at": "2026-10-11 18:48:14",
  "updated_at": "2026-10-11 18:48:14"
}