{
  "id": 214,
  "curve_key": "27798226688830213653489:884868389321601533372341172335287",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-579129722683962784448",
    "-1024153228236700825546996771869"
  ],
  "rank_lower_bound": 14,
  "naive_height": 155.03777755009628,
  "faltings_height": 10.98580287758095,
  "conductor": "80368938829765696602245459305620130547747310",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "73",
    "241",
    "353",
    "171161",
    "279750562966837790136450937"
  ],
  "discriminant": "11977921723624303283051037363414017941731155172995286983285145600",
  "regulator": "48717653573893598.9939311056194054995916665713640284929986128650103",
  "points": [
    [
      "-12938798579711275/561001",
      "106534889369693588972853/420189749"
    ],
    [
      "-9165284686125091/664225",
      "1127734230889137040381743/541343375"
    ],
    [
      "-8432396697289099/3916441",
      "3575196644528288928032739/7750636739"
    ],
    [
      "-1618070764674451/112225",
      "78217983332927259567087/37595375"
    ],
    [
      "-21563290651425/961",
      "24444873195912878927/29791"
    ],
    [
      "-10810436653197/841",
      "50551889024771630845/24389"
    ],
    [
      "-2622368593781/225",
      "6869114780706266423/3375"
    ],
    [
      "-90158812763/9",
      "52438974274566587/27"
    ],
    [
      "20952892100429/625",
      "64874709412740057417/15625"
    ],
    [
      "100146583014325/729",
      "986499899603264866051/19683"
    ],
    [
      "165696032027705/2209",
      "2017400938636708177491/103823"
    ],
    [
      "594013394643365/17161",
      "10154350491931317657147/2248091"
    ],
    [
      "1300448789723773/25281",
      "41247706265133603067519/4019679"
    ],
    [
      "6535886916365997/1681",
      "528382422141653838127321/68921"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥14. 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)=(-12,-3), giving a=(-12,-3,15,16,-27/2,-5/2) with shift t=-1161/88 (cleared: tuple (-24,-6,30,32,-27,-5), t=-1161/44). This family satisfies the Mestre S=0 degree-drop condition identically — it is a union of two depressed cubics with equal sum of squares. Certified rank 14 via 14 independent points from an integer quartic-point search + Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-25 20:54:02",
  "updated_at": "2026-07-01 22:44:38"
}