{
  "id": 209,
  "curve_key": "1421665827432625232798174635095841:-52984092323815152647441895199926527602915169571889",
  "ainvs": [
    "1",
    "0",
    "0",
    "-29618038071513025683295304897830",
    "61324180930341609392295317188644303710246778852"
  ],
  "rank_lower_bound": 12,
  "naive_height": 229.0114121122855,
  "faltings_height": 16.955808744559274,
  "conductor": "28341832953461142174879989692232359137024812210317425503392357305906630092930",
  "bad_primes": [
    "2",
    "5",
    "13",
    "17",
    "23",
    "31",
    "157",
    "6329",
    "694987",
    "462422851227531199",
    "20539201174156553881",
    "210944648998482573373"
  ],
  "discriminant": "38230734935601059553922446496293184814658591376828071969280861189829807789616058154834984960000",
  "regulator": "1760119853264209.7771865074582412243590679211918621886021916407",
  "points": [
    [
      "6103105912463294758/9",
      "-15076912576194046638463529894/27"
    ],
    [
      "2721078832888637224/81",
      "-4432992434214499160854395398/729"
    ],
    [
      "39890447057124135690412/1203409",
      "-7865840853234332855580384931422342/1320139673"
    ],
    [
      "4306516726393068",
      "-116801695188176626466934"
    ],
    [
      "586432636053005156/169",
      "-40007927829283736740469774/2197"
    ],
    [
      "3414095341478124",
      "-480513742457692657782"
    ],
    [
      "491831239431911341/144",
      "4806745636338001659536669/1728"
    ],
    [
      "104843902700829269/16",
      "24671933214190445149476127/64"
    ],
    [
      "845855098895920330636/100489",
      "20357579809465838424373863781554/31855013"
    ],
    [
      "9980933142991084",
      "871779419766022779836298"
    ],
    [
      "13084656463879404",
      "1383467833157765669272458"
    ],
    [
      "7989839137371514742387916/387814249",
      "21864064605954559349916148628421134226/7637226005557"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=5485/16: 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 16:04:29",
  "updated_at": "2026-07-01 22:48:33"
}