{
  "id": 6475,
  "curve_key": "7315353897836761:664850036685140194915",
  "ainvs": [
    "1",
    "1",
    "0",
    "-152403206204932",
    "-769565858610386540"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 109.58625542387338,
  "faltings_height": 7.200428295042944,
  "conductor": "37758135801974184873499473382852152425719842",
  "bad_primes": [
    "2",
    "3",
    "19469",
    "1625749",
    "198820953883898776903491784943147"
  ],
  "discriminant": "226548814811845109240996840297112914554319052",
  "regulator": "25747623610011.21466951017132051774928875840215526523960695118164765300",
  "points": [
    [
      "-7923006/25",
      "861507111362/125"
    ],
    [
      "184652959528",
      "-79347804279794978"
    ],
    [
      "-1223537294/2025",
      "869754895951636/91125"
    ],
    [
      "87971882170/529",
      "-26021522491762660/12167"
    ],
    [
      "604080694238/5041",
      "-467030650735326620/357911"
    ],
    [
      "-3965830379/400",
      "185193024462959/8000"
    ],
    [
      "-4027689497/484",
      "-279899357015039/10648"
    ],
    [
      "84854789320/2209",
      "23404627402912310/103823"
    ],
    [
      "52436459781/625",
      "11876065600105354/15625"
    ],
    [
      "-11694671",
      "13500676894"
    ],
    [
      "-8311117",
      "26305938185"
    ],
    [
      "-7843986",
      "26688251542"
    ],
    [
      "-6857864",
      "26870947654"
    ],
    [
      "-3950786",
      "23232908092"
    ],
    [
      "-1152518",
      "13166706562"
    ],
    [
      "15321016",
      "35497479334"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,10,45,69,71,80,88,94,100,111: 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": "2026-11-04 02:11:09",
  "updated_at": "2026-11-04 02:11:09"
}