{
  "id": 243,
  "curve_key": "37977539895016519929470555599561:-252010427891277533784631384743578099062204450309",
  "ainvs": [
    "1",
    "0",
    "1",
    "-791198747812844165197303241658",
    "291678735985274857428612896086571996361540568"
  ],
  "rank_lower_bound": 20,
  "naive_height": 218.29159930381655,
  "faltings_height": 16.146949249509756,
  "conductor": "123989445321322909065995690886185865732040186216447789402341303782551770",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "19",
    "23",
    "29",
    "31",
    "59",
    "1657",
    "204191283231710516050941879612935200616637742318196045393"
  ],
  "discriminant": "-5054684863451019643038749147096969353867668542170076589556829874258223258043978504934937500",
  "regulator": "26922583187358235172.6849204969274253584348869249045696391195036218961397153261",
  "points": [
    [
      "436876547159399",
      "5422665630197378101065"
    ],
    [
      "375990611505449",
      "-6881048738091909428565"
    ],
    [
      "475838366133299",
      "-4789192526536875389535"
    ],
    [
      "309626103514289",
      "8739924684576584646255"
    ],
    [
      "579770808085271/4",
      "-107344757768733870289755/8"
    ],
    [
      "-151670202942496",
      "20203737650008769711175"
    ],
    [
      "474836608215689",
      "4801033336095140780355"
    ],
    [
      "1399900845934769",
      "43903256069991280960095"
    ],
    [
      "413316049389764",
      "5938892007725126484555"
    ],
    [
      "-491678658233761",
      "23702995629264462353655"
    ],
    [
      "207182831653769",
      "-11689705199855651038365"
    ],
    [
      "2826878413762139",
      "-143684810483260713380445"
    ],
    [
      "972792119473619",
      "21037676494215690500235"
    ],
    [
      "733163671616639",
      "-10280918119880512036695"
    ],
    [
      "1159271608488944",
      "-30535585821516332183880"
    ],
    [
      "584393010154124",
      "5374644645235395266835"
    ],
    [
      "1001929050655889",
      "-22466688030826232997945"
    ],
    [
      "382267730323742",
      "6714852110077627182165"
    ],
    [
      "467508117994889",
      "4895637428325170187555"
    ],
    [
      "3327907052188819",
      "-185782396222315328318005"
    ]
  ],
  "submitter": "David Renshaw",
  "commentary": "New curve of rank >= 20 and naive height 218.2916, beating the previous rank-20 record 223.3165 (#92). Found as the specialization T = 3895/6 of the Mestre/Fermigier rank-12 quartic family on the rational Mestre sextuple mestre_ais(u = -7/2, v = 3/2) (roots {-1455/4, 2955/4, 1437/2, -1149/4, -1851/4, -687/2}): q(x,T) = p6(x-T) p6(x+T) = g^2 - r with deg r = 4, curve = Jacobian of y^2 = r(x). Candidate surfaced by a staged Mestre-Nagao sieve (M=1500 stage 1 over T = m/n, m <= 5000, n <= 20; M=6000 rescore ranked it #1 of 3135 candidates under the rank-20 height bar, score 103.65 vs 98.36 for Fermigier's rank-22 curve in the same normalization). The 20 independent points were assembled from hyperellratpoints on the integral quartic model to height 1e7 plus the 12 injected family base points x = ai +/- T and an integer sweep ellratpoints(E, [1e12, 1]); independence certified by positive-definite Neron-Tate Gram matrix (eigenvalue margin 1e-7 at 90 digits).",
  "created_at": "2026-07-02 02:02:39",
  "updated_at": "2026-07-02 02:02:39"
}