{
  "id": 6491,
  "curve_key": "17126015686220941545736656:-71265367470214450835199493605303203520",
  "ainvs": [
    "0",
    "0",
    "0",
    "-356791993462936282202847",
    "82483064201637095874073487969100930"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 174.31894765494096,
  "faltings_height": 12.366256119579894,
  "conductor": "298924201245331810645987537174751475783039180013627808551556424",
  "bad_primes": [
    "2",
    "3",
    "11",
    "19",
    "199",
    "2008121",
    "854639069945171972401",
    "58164312337319119233735288847"
  ],
  "discriminant": "-32225186328753991112408277771182149727125855533174090528732494133673728",
  "regulator": "3507692373111666049.05211655466379882229932763024426543841236644464273655817686",
  "points": [
    [
      "745450158733221/25",
      "-20348923391510537211894/125"
    ],
    [
      "-637000077783",
      "-226461131552584962"
    ],
    [
      "76642590258709/25",
      "-659092633390875227302/125"
    ],
    [
      "1263693043258",
      "-1284377022045453446"
    ],
    [
      "733648488058041/625",
      "-17685206865672831825714/15625"
    ],
    [
      "12112781780865/16",
      "-31759492349514544065/64"
    ],
    [
      "59900990254485/169",
      "51490096232399835750/2197"
    ],
    [
      "194662397318457/256",
      "2051470070093291470227/4096"
    ],
    [
      "1099770464765013/1369",
      "28397098809496139789286/50653"
    ],
    [
      "-4135787568354399/6241",
      "-82367411936321749691718/493039"
    ],
    [
      "84497201632714785/299209",
      "10664919448007183922002310/163667323"
    ],
    [
      "54795862870663/289",
      "-722892703935253974056/4913"
    ],
    [
      "-10021817220718719/16129",
      "-519354666496500416635482/2048383"
    ],
    [
      "-405209548263",
      "400656080541235662"
    ],
    [
      "-2633064700887/4",
      "1433562953691497571/8"
    ],
    [
      "1042752383635398465/3389281",
      "267197925281990796030107610/6239666321"
    ],
    [
      "92006595532323/289",
      "-167434103348891315856/4913"
    ],
    [
      "5208342249/4",
      "-2291109672881692611/8"
    ],
    [
      "-123195576367883/361",
      "-2781908639837378611642/6859"
    ],
    [
      "-2799043108184754531/8508889",
      "-10059316129093280274516582126/24820429213"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,15,24,25,32,52,128,148,195: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 20: the 20 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:53:11",
  "updated_at": "2026-11-04 04:53:11"
}