{
  "id": 6489,
  "curve_key": "247045416135024140043753:-127158349403136802073307367634731125",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-5146779502813002917578",
    "147174015514176512090179370677000"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 161.66148244857905,
  "faltings_height": 11.392686166291291,
  "conductor": "31737968087921775307522975461902370443739390331195531141429586",
  "bad_primes": [
    "2",
    "37",
    "269",
    "2081",
    "417581948294786754772603031",
    "1834768700946805932206043071"
  ],
  "discriminant": "-631775992758170859271552349544628586053076303932778242901297338916",
  "regulator": "20254320983398498603.7064329186707485602436402149914632427818981608835839898442",
  "points": [
    [
      "9289622161606/625",
      "134373998654294956554/15625"
    ],
    [
      "695051778182",
      "-576496060284340614"
    ],
    [
      "2232136032572/169",
      "19833912196471111692/2197"
    ],
    [
      "1003478511477/16",
      "-539573957469023897/64"
    ],
    [
      "146077480847950/2401",
      "-905558115117645069270/117649"
    ],
    [
      "-12598166070139/400",
      "-133394145238281344619/8000"
    ],
    [
      "137159916939394/2025",
      "952732291645556476082/91125"
    ],
    [
      "100228150955951/1225",
      "709437738126138665496/42875"
    ],
    [
      "62913357737246/625",
      "398073438945696488344/15625"
    ],
    [
      "43095547097410700/1142761",
      "3165030610212200139034080/1221611509"
    ],
    [
      "7359188560950/361",
      "-48851207876006514370/6859"
    ],
    [
      "-21423360006",
      "15735411970720588"
    ],
    [
      "-99771831139956/1225",
      "218978356490301610208/42875"
    ],
    [
      "5490544241263/169",
      "8294828093497417224/2197"
    ],
    [
      "-2719981932293/36",
      "2210482933970425727/216"
    ],
    [
      "558025827370437676/16152361",
      "211340326478193103124861484/64916338859"
    ],
    [
      "7185845623010063/208849",
      "313969641521172930966288/95443993"
    ],
    [
      "8432497473311521/326041",
      "1042571697844854657724671/186169411"
    ],
    [
      "11774270481040591/514089",
      "2369116584573373495543784/368601813"
    ],
    [
      "137728761388924066/4748041",
      "48842040512737714558111662/10345981339"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,25,26,48,49,80,135,140,150,175: 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 04:52:57",
  "updated_at": "2026-11-04 04:52:57"
}