{
  "id": 239,
  "curve_key": "1073250816532434693568:-35158133924832719085472876696064",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-22359392011092389449",
    "40692284642676185167439218585"
  ],
  "rank_lower_bound": 15,
  "naive_height": 145.27493742540233,
  "faltings_height": 9.786107191203211,
  "conductor": "493331968419716917139720664887174469134762256768",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "29",
    "4868091091",
    "13751713379",
    "7271987713848853806487"
  ],
  "discriminant": "85738128226224787181909938997870961494015365496541683712",
  "regulator": "72348684025139.813690956329432821964132838122276072356655182051278552",
  "points": [
    [
      "10851060085/4",
      "-258818176155/8"
    ],
    [
      "7558450841269/289",
      "20461735427020481412/4913"
    ],
    [
      "36592676846697/1024",
      "-219509068333802137163/32768"
    ],
    [
      "12690550357/4",
      "-328743630743307/8"
    ],
    [
      "-21288072143/16",
      "-16699737845264319/64"
    ],
    [
      "1634480901",
      "-92265088908044"
    ],
    [
      "-2111327789",
      "-280158194286516"
    ],
    [
      "19227744004225597/3261636",
      "-1986693395508881051828039/5890514616"
    ],
    [
      "40409079012536/19321",
      "-148973071535977430301/2685619"
    ],
    [
      "208147940321649/76729",
      "390225018276560012/21253933"
    ],
    [
      "-134774415986415/139129",
      "-12863588512365650166380/51895117"
    ],
    [
      "1921166487515869/822649",
      "-25976048739534448334052/746142643"
    ],
    [
      "269290852319401/103041",
      "-342687181344661622924/33076161"
    ],
    [
      "60150234401830765/21353641",
      "-764007110263969721150820/98675175061"
    ],
    [
      "4103100327",
      "-134264379870296"
    ]
  ],
  "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)=(1,-3), cleared tuple [20, -60, 40, 125, 3, -128], shift t=213. Two depressed cubics with equal product (satisfies Mestre 2e5=e2e3 identically). Certified rank 15 via 15 independent points (quartic point search + Néron–Tate height-pairing matrix). Smallest-conductor rank-15 of the family.",
  "created_at": "2026-06-26 15:03:44",
  "updated_at": "2026-07-01 22:42:37"
}