{
  "id": 6493,
  "curve_key": "12875328279575444957876841:-46393364043745311560390913089668379445",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-268236005824488436622434",
    "53696023198846354788945604851827344"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 173.4604395609829,
  "faltings_height": 12.283469125187668,
  "conductor": "15141241539563545332571561817468094769657291789078778794197996638938",
  "bad_primes": [
    "2",
    "7",
    "97829917669660331537",
    "11055076796424360976352235184081525568192325691"
  ],
  "discriminant": "-10386891696140592098144091406783113011984902167308042252819825694311468",
  "regulator": "43160327638106075202.2980752951799132595670937171201881568299089958658725751524",
  "points": [
    [
      "3354307890078",
      "-6074089867875408098"
    ],
    [
      "129586837313350",
      "-1475155802140200997762"
    ],
    [
      "-438095411359694/1521",
      "-19409248468372185086054/59319"
    ],
    [
      "50400792824241/256",
      "378018853391388608239/4096"
    ],
    [
      "844584534257246/4225",
      "24660326000631354819486/274625"
    ],
    [
      "742101740308864/3249",
      "-12206930720045775434188/185193"
    ],
    [
      "242799558515961/961",
      "-1349876203969024866626/29791"
    ],
    [
      "1767583330903774/5625",
      "8805356800071478014332/421875"
    ],
    [
      "7192427499933378/22801",
      "74704263117277885479802/3442951"
    ],
    [
      "1420470284519/4",
      "-454246799007164085/8"
    ],
    [
      "22838018146410413/38416",
      "-2432146598318551059827645/7529536"
    ],
    [
      "45656578853",
      "-203824597021926098"
    ],
    [
      "-25374348664651262478/44448889",
      "-42722999741305872468806617150/296340742963"
    ],
    [
      "-8195469404745557/16641",
      "-552951981342905655733972/2146689"
    ],
    [
      "1047472748285101/400",
      "-33283153030022429918511/8000"
    ],
    [
      "884800433571092933/737881",
      "-764877012586513088440317630/633839779"
    ],
    [
      "3844135683097",
      "-7471869888989119227"
    ],
    [
      "8605433269055",
      "-25199363510831912847"
    ],
    [
      "8860738291627",
      "-26331682391635300567"
    ],
    [
      "109670092635502007/58564",
      "-35056414517872720818061345/14172488"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,39,64,65,114,124,150,151,168: 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 04:53:25",
  "updated_at": "2026-11-04 04:53:25"
}