{
  "id": 6183,
  "curve_key": "3325629010137904561:-6367431792219660605785244905",
  "ainvs": [
    "1",
    "0",
    "0",
    "-69283937711206345",
    "7369712716702797558539801"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.04198745358298,
  "faltings_height": 8.606526351235953,
  "conductor": "29109434091169948293553656515843844039762",
  "bad_primes": [
    "2",
    "3",
    "29",
    "595207",
    "281071288182287246899399535819209"
  ],
  "discriminant": "-2177875148181136300520044234649013726059485978165248",
  "regulator": "714489547613882627.7294154584799142406449441296494732053126403111618102",
  "points": [
    [
      "137806634",
      "662476613075"
    ],
    [
      "-98500696588726/398161",
      "769056184699938185381/251239591"
    ],
    [
      "-8484971345590/27889",
      "2497723768028482453/4657463"
    ],
    [
      "-1697661276106/7225",
      "2006712204679946627/614125"
    ],
    [
      "-121200481744/1225",
      "156105376646252599/42875"
    ],
    [
      "-1482783158/9",
      "102148114970801/27"
    ],
    [
      "116998923713/1024",
      "31854481872901327/32768"
    ],
    [
      "62886740218/1089",
      "67817413720772483/35937"
    ],
    [
      "149038334858/361",
      "48076933907749121/6859"
    ],
    [
      "205212773210/529",
      "75854051441531893/12167"
    ],
    [
      "227969309690/841",
      "71130475801460431/24389"
    ],
    [
      "10218350234321/53824",
      "12848023805545591883/12487168"
    ],
    [
      "4774481178554/26569",
      "3680263807068744737/4330747"
    ],
    [
      "6903318620954/7921",
      "17396155266375897595/704969"
    ],
    [
      "26811449679512/218089",
      "85820225460641418079/101847563"
    ],
    [
      "51515312095103834/134351281",
      "9494971074153054868333285/1557265698071"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,11,52,54,67,71,75,128,140: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:41:08",
  "updated_at": "2026-10-30 14:41:08"
}