{
  "id": 316,
  "curve_key": "56917675259131909898858792689:-13573416748305432704960324874308037479316937",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1185784901231914789559558181",
    "15709973088316374222740088167179617119361"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "naive_height": 198.63421032293056,
  "faltings_height": 14.295376677825272,
  "conductor": "350481036484526026323524826173099607322997042184041024971581818910",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "10667",
    "3889864799",
    "17205539371",
    "84895016131819",
    "2118237190297826516065951"
  ],
  "discriminant": "89176446007892832942170673747379791745757428321235958497602360325335450136985600",
  "regulator": "92302900417207424.6618806031761127297795276055382073375398310158206493746",
  "points": [
    [
      "4595707826225396946",
      "-9852095564070173687393395353"
    ],
    [
      "27330565789098",
      "60964201325242834047"
    ],
    [
      "19528601652684",
      "895714120900637703"
    ],
    [
      "144365726817442/9",
      "771518233290848301245/27"
    ],
    [
      "20214959284656",
      "336313418519618847"
    ],
    [
      "239787319553506/9",
      "-1486143331484671369859/27"
    ],
    [
      "-39296468065374",
      "-40311743329885274793"
    ],
    [
      "3905575201413379/324",
      "-328242264546675851387821/5832"
    ],
    [
      "18061975170018",
      "-13594241571089606505"
    ],
    [
      "-76369704710221/4",
      "1417375645728035493331/8"
    ],
    [
      "12275980540434",
      "54802250006833328103"
    ],
    [
      "8816052662491586/529",
      "-292243788677002214771279/12167"
    ],
    [
      "18370790904846",
      "-11227234400360350653"
    ],
    [
      "349519770477602876034/11662225",
      "-3353820769946805438405452648607/39826498375"
    ],
    [
      "46544298010424407728000/2381147209",
      "-27245787719894335000213207371387/116192840357573"
    ],
    [
      "13182733475466182730/124609",
      "-45591976984664375625274815753/43986977"
    ],
    [
      "3902300440344891/100",
      "-169885612449259904662569/1000"
    ]
  ],
  "submitter": "Jack Cheng",
  "commentary": "A new canonical curve from the Mestre-Fermigier family on [348,-600,-216,492,876,-900] at the fresh shift t=2/9. Completing p6(x-t)p6(x+t) gives, after square normalization, Y^2=450555686361*x^4-8079008597280*x^3-158594291989460424*x^2+3160507498316956800*x+13015610394173754277264. Its 11 generic construction directions are enlarged by six independent points found in hyperellratpoints(Q,[1000000,2000]), with primitive-quartic sources (5618/9,140122690848), (9410/9,610042859808), (-4258/27,848258843264/9), (-33127/45,6138313886721/25), (7904/45,2379994816884/25), and (-4034/81,9044700913696/81); these map respectively to the final six displayed witnesses. The ICARM exact Cremona/Brumer verifier certifies all 17 points: F_2 rank 17, torsion rank 0, no halvings, using 18 good primes. No parity or BSD assumption is used.",
  "created_at": "2026-08-24 07:32:42",
  "updated_at": "2026-08-24 07:32:42"
}