{
  "id": 6468,
  "curve_key": "705651564247095343734801:-577446373938656732425328376995489145",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-14701074255147819661142",
    "668340710580824264426435539400805"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 164.74022560043676,
  "faltings_height": 11.636336540670204,
  "conductor": "10883578061541468914591934184166771949462813702546719742",
  "bad_primes": [
    "2",
    "3",
    "13",
    "109",
    "6113262985406879",
    "69800134932133388125350454276890833"
  ],
  "discriminant": "10376585390636390892311839439864587208459077840846239496217554747392",
  "regulator": "311780962404780454.8649661909433819425125693159743154142188947322967534526712",
  "points": [
    [
      "8750257949329/9",
      "-25691425140291627191/27"
    ],
    [
      "-9308892567",
      "28361682405949067"
    ],
    [
      "5023280920099/36",
      "-7888601304755069825/216"
    ],
    [
      "86333421113173/1089",
      "-38203885267757926141/35937"
    ],
    [
      "133858150257221/1681",
      "111625494779226529227/68921"
    ],
    [
      "148430729746533/1849",
      "186724771449804982089/79507"
    ],
    [
      "1456095287621/16",
      "587175247316376063/64"
    ],
    [
      "2586839256461/25",
      "1996227473963682519/125"
    ],
    [
      "1092757924693/9",
      "700614086616476869/27"
    ],
    [
      "-139502853847",
      "2076373110594699"
    ],
    [
      "-5723648407",
      "27428035661058699"
    ],
    [
      "84306542697",
      "5306611297883915"
    ],
    [
      "99987132713",
      "14072493453095499"
    ],
    [
      "100709274281",
      "14464931409351051"
    ],
    [
      "268573850537",
      "126857117233987467"
    ],
    [
      "274366825577",
      "131485252438894347"
    ],
    [
      "297281102261",
      "150234733551381927"
    ],
    [
      "2796088654073/49",
      "1339925927940728925/343"
    ],
    [
      "9695646975377/169",
      "8149471508755875303/2197"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,12,33,41,43,56,65,78,114: 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 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:10:16",
  "updated_at": "2026-11-04 02:10:16"
}