{
  "id": 6169,
  "curve_key": "34301815933933015209:-144579123625920046840835089173",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-714621165290271150",
    "167336948819766456647608400"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 134.9451899080307,
  "faltings_height": 9.270599255843134,
  "conductor": "82254192485037013616681222384133359549202218288370",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "37649",
    "508563435139",
    "3671754232125353251062853117451"
  ],
  "discriminant": "11259776409276716793987492532164015588690291661494969300",
  "regulator": "626953721580525584073.19230068903771176187624561108197589887931236168825618",
  "points": [
    [
      "28784201080",
      "4881397117944400"
    ],
    [
      "36718095670",
      "7034037760791730"
    ],
    [
      "-15152176309194602/29822521",
      "3254256301875056041297384/162860787181"
    ],
    [
      "-688700247645850/966289",
      "16848435590989951213790/949862087"
    ],
    [
      "-660091906607513/822649",
      "11170769331411775371127/746142643"
    ],
    [
      "-177234846840992/192721",
      "578503324619372653396/84604519"
    ],
    [
      "-15780709938545/143641",
      "851289018800817011575/54439939"
    ],
    [
      "55556569399/49",
      "9789591733360132/343"
    ],
    [
      "56435626825/49",
      "10129008348890215/343"
    ],
    [
      "78154918999/25",
      "21097028752838708/125"
    ],
    [
      "90178528705/121",
      "9288067020375395/1331"
    ],
    [
      "6860570220394/529",
      "17932075426803074918/12167"
    ],
    [
      "16089980834680/67081",
      "54187705599734790880/17373979"
    ],
    [
      "22513442300512/29929",
      "38545163894971821476/5177717"
    ],
    [
      "63069846981022/69169",
      "301024104703960595408/18191447"
    ],
    [
      "83571470757286/121801",
      "6997958797475690296/42508549"
    ],
    [
      "1323172603979995/1885129",
      "8793603035155839409865/2588282117"
    ],
    [
      "7948092878466086/5958481",
      "579509529581124964567394/14544652121"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,14,17,21,25,74,92,106,121: 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": "2026-10-30 14:39:25",
  "updated_at": "2026-10-30 14:39:25"
}