{
  "id": 212,
  "curve_key": "10634696675535523446376517660944:-30573625125649705614695987633207721066059380672",
  "ainvs": [
    "0",
    "1",
    "0",
    "-221556180740323405132844117936",
    "35386140191724122461245294467670188433973860"
  ],
  "rank_lower_bound": 11,
  "naive_height": 214.32502415017666,
  "faltings_height": 15.842747841238426,
  "conductor": "1103561624055499058867562340698878392772504928025988266715523317532246643920",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "31",
    "157",
    "670606297099",
    "1575838430456954508271967",
    "81274068710384465721193186106423"
  ],
  "discriminant": "155094517685876658621636759147912906994801674312562010974276152381499783783841112416793600",
  "regulator": "15079211501874.5923514849165940580973990610604958433553929196",
  "points": [
    [
      "1439725582641991678/9",
      "-1727498601505665033867479240/27"
    ],
    [
      "1609812018760123810/81",
      "-2041934892838682216734870460/729"
    ],
    [
      "382889599831597753/144",
      "-233409324592028354395076875/1728"
    ],
    [
      "1461300830275451952/841",
      "-1706610970573605828131905230/24389"
    ],
    [
      "3003181201931493/4",
      "-136765210570118860311615/8"
    ],
    [
      "4642447936168800893203/13344409",
      "31359710078558692581991890119040/48747126077"
    ],
    [
      "24321114180952583584546/63696361",
      "1291845898306026112352226583813164/508360657141"
    ],
    [
      "38661152893172916450454/78234025",
      "4722711530104732719979077042193992/691979951125"
    ],
    [
      "721344004004635",
      "15840139419502659908460"
    ],
    [
      "779580898158616",
      "18342645394695663676806"
    ],
    [
      "16344491930375377/16",
      "-1893202461186603241978305/64"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=14612/9: 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 11 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 16:04:44",
  "updated_at": "2026-07-01 22:49:33"
}