{
  "id": 6129,
  "curve_key": "16217902727614041420801:-647578738239802113470192715820545",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-337872973491959196267",
    "749512428602757726556083220155"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 153.42120807472588,
  "faltings_height": 10.847079886211393,
  "conductor": "55035751703278323680294108487782527741865489894062",
  "bad_primes": [
    "2",
    "7",
    "13",
    "23",
    "54611491665678687743",
    "240747418844039285759013869"
  ],
  "discriminant": "2225856791490090803129792984855460575238958953485740217386811392",
  "regulator": "5281701513166674130.7204750743036580891528468651164509782738752883178188784",
  "points": [
    [
      "-18528264265",
      "805628575796110"
    ],
    [
      "-11524538773",
      "1764293455553450"
    ],
    [
      "17498884123",
      "442102402103274"
    ],
    [
      "49360784767",
      "10214609492812602"
    ],
    [
      "1593154542315/4",
      "2008750985085040857/8"
    ],
    [
      "784894354887",
      "695180808913910270"
    ],
    [
      "-727989512064899/52441",
      "19967712224021608989828/12008989"
    ],
    [
      "-79781481273733/5041",
      "522682751579736273862/357911"
    ],
    [
      "-73073409534545/17161",
      "3266324369304531834170/2248091"
    ],
    [
      "-22411305980861/7225",
      "816514298624802752418/614125"
    ],
    [
      "-1894663561273/121",
      "1974588034260886670/1331"
    ],
    [
      "-1530711723653/81",
      "452778609176253818/729"
    ],
    [
      "190158042067/9",
      "47098912690667198/27"
    ],
    [
      "1326143585431/9",
      "1515413573203548190/27"
    ],
    [
      "2530987583859/25",
      "3961115940750459138/125"
    ],
    [
      "10653478494043/5041",
      "75841378842677664422/357911"
    ],
    [
      "1522814754009843/85849",
      "14613314378632321309570/25153757"
    ],
    [
      "2644651451997527/1369",
      "135998109120634475802622/50653"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,13,53,66,86,89,96,107,113,135: 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": "2026-10-29 12:55:48",
  "updated_at": "2026-10-29 12:55:48"
}