{
  "id": 238,
  "curve_key": "1232388808397940921824433:-1368984847784535468478339539721326441",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-25674766841623769204676",
    "1584473203460742150597373002163815"
  ],
  "rank_lower_bound": 15,
  "naive_height": 166.41426565188104,
  "faltings_height": 11.625099081975636,
  "conductor": "95954989391736297575743447520151210094560387026536946",
  "bad_primes": [
    "2",
    "7",
    "13",
    "23",
    "67",
    "614071",
    "2845615572511",
    "195793702136100389221771138343"
  ],
  "discriminant": "-1382730624930385367184147512181586170312381345872576399281800347648",
  "regulator": "91113336322005.277313711589725666999350825220758797042718744371374129",
  "points": [
    [
      "4686108375477/49",
      "-664183300877107717/343"
    ],
    [
      "131018426453",
      "21671238900850445"
    ],
    [
      "721984601739957/5329",
      "-9471872484708889421835/389017"
    ],
    [
      "2694575359197861/2209",
      "-138723138874027700845101/103823"
    ],
    [
      "190686676611340509/2247001",
      "-13804565079891464631408393/3368254499"
    ],
    [
      "14987765522285/169",
      "-4920895085425106975/2197"
    ],
    [
      "24227790657649/225",
      "-27903398879047177237/3375"
    ],
    [
      "3819910387281869/42025",
      "-11330999487134645156863/8615125"
    ],
    [
      "77611481861",
      "-7701501794569091"
    ],
    [
      "2638222669560109/26569",
      "-16264401587478186117297/4330747"
    ],
    [
      "271036990440985/289",
      "-4400877261188912276575/4913"
    ],
    [
      "229833258657625885/2627641",
      "-12003197530901784628769351/4259406061"
    ],
    [
      "16035175610211357/35344",
      "1918001943185220915702165/6644672"
    ],
    [
      "1552119729577773/17161",
      "-3324258507528512596305/2248091"
    ],
    [
      "121581860292421693/961",
      "-42393848878672419779626805/29791"
    ]
  ],
  "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)=(-7,-1), cleared tuple [-210, -30, 240, 189, -125, -64], shift t=1791/4. 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 06:57:35",
  "updated_at": "2026-07-01 22:42:38"
}