{
  "id": 205,
  "curve_key": "865501542262379153750461947664:-810356568344230487580803832135872788497253312",
  "ainvs": [
    "0",
    "1",
    "0",
    "-18031282130466232369801290576",
    "937912694842853350198553539190914682552740"
  ],
  "rank_lower_bound": 12,
  "naive_height": 206.8120965287715,
  "faltings_height": 15.080140699542088,
  "conductor": "5883556108705604864069301892047856053057376997561594085263469332125323440",
  "bad_primes": [
    "2",
    "5",
    "13",
    "17",
    "19",
    "23",
    "157",
    "241",
    "617",
    "41665714723",
    "46051797406879788477331351041654345220422732481"
  ],
  "discriminant": "-4824473724726869454399342069094860042611338540998910914446745354020291555955475532800",
  "regulator": "23816794754349.050264022329195680584976251521649799131276940920",
  "points": [
    [
      "1019510314870079278/9",
      "-1029407033233963549733458760/27"
    ],
    [
      "898379318706026386/81",
      "-851447619821221046090639324/729"
    ],
    [
      "65554377206988649/144",
      "-16124573198339556102902467/1728"
    ],
    [
      "89943481345135872/841",
      "-11766652141274314098778430/24389"
    ],
    [
      "558225231481187845883/7360369",
      "-1625706497883799859381407016320/19968681097"
    ],
    [
      "288028626599061/4",
      "-908295957363279970007/8"
    ],
    [
      "36286970506582",
      "-575667222416485993560"
    ],
    [
      "17527214855180285974742/58507201",
      "2119063633757101595805966005518120/447521580449"
    ],
    [
      "855715812309179",
      "24740771596287139505996"
    ],
    [
      "987441988220392",
      "30755977676121784722870"
    ],
    [
      "41176586031416048557327/31348801",
      "8313519254200782912639963937076910/175521936799"
    ],
    [
      "17435634703211710010126/200647225",
      "-468872888856202121848899501476824/2842167942125"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=10852/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-25 16:04:09",
  "updated_at": "2026-07-01 22:46:38"
}