{
  "id": 6484,
  "curve_key": "3652514315345659426584316641:-222192209000963033860196203370334258770385",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-76094048236367904720506597",
    "257166908565948460850249142173326259485"
  ],
  "rank_lower_bound": 21,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 190.40872288071324,
  "faltings_height": 13.714183148046054,
  "conductor": "80043266134234367016571902619913942762718535666698465277063660482",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "739",
    "23669",
    "9148957",
    "94320931",
    "447911121627965663",
    "7227948420235368763549"
  ],
  "discriminant": "-371349885848318138853423248205817314992901319998717759199621733582683667693568",
  "regulator": "24318006525217584197.443925281143923916306997604557935069655517884153158194673449",
  "points": [
    [
      "484758981684847/81",
      "-2926735447221230093792/729"
    ],
    [
      "27034819108351",
      "-134012049221092391968"
    ],
    [
      "2868790678481911/1225",
      "410813581173239511792736/42875"
    ],
    [
      "4367085119428855/841",
      "34920850589635339511200/24389"
    ],
    [
      "50403606201343/9",
      "69696793253977085864/27"
    ],
    [
      "150148277789131/25",
      "512232639254113176556/125"
    ],
    [
      "2612618903974943/289",
      "86233196424036677554976/4913"
    ],
    [
      "125629053739446295/11881",
      "32628758731694503179098336/1295029"
    ],
    [
      "6962901864412207/361",
      "529740805024376064582968/6859"
    ],
    [
      "8428299116447495/1849",
      "-178141974229909198256800/79507"
    ],
    [
      "8030911701280207/1681",
      "-112659545942861037912320/68921"
    ],
    [
      "-25152543985450697/2809",
      "-2211159160916640466595360/148877"
    ],
    [
      "5671558582374379/1225",
      "-86880485033486292060128/42875"
    ],
    [
      "333052872712116623/69169",
      "-28215666729444722588117536/18191447"
    ],
    [
      "-501244254581301457/49729",
      "3729234441511289914905040/11089567"
    ],
    [
      "389530233356458986775/97397161",
      "-3940688461625029024533007787360/961212581909"
    ],
    [
      "7573883346403",
      "-10738043819179201156"
    ],
    [
      "79731868546936719487/16008001",
      "-84078326643798811118900745440/64048012001"
    ],
    [
      "8432802058176751/1681",
      "-89403237059480951792288/68921"
    ],
    [
      "855718480382945/169",
      "-2854306668476559083900/2197"
    ],
    [
      "18561847632959558287/3690241",
      "-9181173363333663823533624688/7088952961"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,27,35,58,72,80,102,109,152,193: 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 >= 21: the 21 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 04:52:20",
  "updated_at": "2026-11-04 04:52:20"
}