{
  "id": 6470,
  "curve_key": "501870675127443055056:-8378466932960198486331485296320",
  "ainvs": [
    "0",
    "0",
    "0",
    "-10455639065155063647",
    "9697299690926155655476256130"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 142.9946224233809,
  "faltings_height": 9.936769892728167,
  "conductor": "196490377341413160535884163143295278957797085473064",
  "bad_primes": [
    "2",
    "3",
    "13",
    "17",
    "18223",
    "133319",
    "5082818406145693511698600758193219481"
  ],
  "discriminant": "32528677015805314853490471516151335037190364998989091004672",
  "regulator": "14880942986461649.175758522517106470479333431545035722155401035956421504027",
  "points": [
    [
      "90483665277",
      "-27200734802524062"
    ],
    [
      "1955089820769/25",
      "-2731385044873494522/125"
    ],
    [
      "-1851232197894/625",
      "1893168056381110254/15625"
    ],
    [
      "389786938809/961",
      "2214007102177261326/29791"
    ],
    [
      "732391197",
      "49320726099138"
    ],
    [
      "8061614410/9",
      "875109714573070/27"
    ],
    [
      "28831759747081/11025",
      "17897401240291914146/1157625"
    ],
    [
      "3055234568221/729",
      "3911433546381271858/19683"
    ],
    [
      "45979877742693/9409",
      "250450664020326036774/912673"
    ],
    [
      "-3216414183",
      "100260315501138"
    ],
    [
      "-1278397251",
      "144825674177826"
    ],
    [
      "-221094219",
      "109536173805942"
    ],
    [
      "58689894",
      "95309293880736"
    ],
    [
      "3149796681",
      "89520866607042"
    ],
    [
      "6724923558",
      "493473285050904"
    ],
    [
      "9473404389",
      "872262105675354"
    ],
    [
      "16495780633/4",
      "1532880960496379/8"
    ],
    [
      "495053357247/121",
      "250444984066338672/1331"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,2,20,31,34,36,63,81,97,116: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:10:32",
  "updated_at": "2026-11-04 02:10:32"
}