{
  "id": 237,
  "curve_key": "79278735968380698064:-786858559855956143215675588288",
  "ainvs": [
    "0",
    "1",
    "0",
    "-1651640332674597876",
    "910715925208661721534017824"
  ],
  "rank_lower_bound": 15,
  "naive_height": 137.67569204492898,
  "faltings_height": 9.446902766047197,
  "conductor": "5919531324154970229554784141330370952904285996920",
  "bad_primes": [
    "2",
    "5",
    "13",
    "19",
    "23",
    "26049688981495204319463052901471444080726483"
  ],
  "discriminant": "-69948023501250722618129515710347501757574437396885241600",
  "regulator": "111257299159024.59816588442634390170741817967774122122164844181174948",
  "points": [
    [
      "772668450",
      "9789861575182"
    ],
    [
      "682982520231464/143641",
      "-17263025154188599268920/54439939"
    ],
    [
      "2306049782",
      "96774035268922"
    ],
    [
      "150596914681/64",
      "51336000912478035/512"
    ],
    [
      "681402367466/49",
      "560166370225180550/343"
    ],
    [
      "2852440687",
      "139313169639822"
    ],
    [
      "18862882604/25",
      "1212361838592208/125"
    ],
    [
      "5397415438786/23409",
      "83394215197427708990/3581577"
    ],
    [
      "78607545570",
      "22036301809608398"
    ],
    [
      "28772514445/36",
      "2172923165525957/216"
    ],
    [
      "-87485153086/841",
      "802024535920980970/24389"
    ],
    [
      "17438427206/25",
      "1237612245267346/125"
    ],
    [
      "1043724563974/441",
      "937999906284868750/9261"
    ],
    [
      "16728707607069638/165649",
      "2163508515105097434676454/67419143"
    ],
    [
      "390658744",
      "18030759283680"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥15. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-10,-1), cleared tuple [-840, -84, 924, 640, -343, -297], shift t=784/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 15 via 15 independent points from a quartic point search + Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-26 06:57:30",
  "updated_at": "2026-07-01 22:41:39"
}