{
  "id": 6492,
  "curve_key": "11419457701775537800517481:-54649480672139547509925585749698600629",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-237905368786990370844114",
    "63251713740961730404610619988195520"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 173.78800611913192,
  "faltings_height": 12.506284208391,
  "conductor": "26334328961446354492261518081380917681712112753779724544284319830",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "313",
    "5580100628223259",
    "12886946520483517602520702479333533488544497"
  ],
  "discriminant": "-866564012387594102510980579365440822463839210304064060785355899405937500",
  "regulator": "9956358436237454257.54895726260377887093717407568272270911471074905871220310542",
  "points": [
    [
      "27277259374/49",
      "-86173776996088395734/343"
    ],
    [
      "129397655757086",
      "-1471927952350318832708"
    ],
    [
      "109766581793344/1369",
      "-10708278553427777609864/50653"
    ],
    [
      "10277752054264264/6889",
      "-995138151079342935436156/571787"
    ],
    [
      "627655569844291/441",
      "-14954948984103013116043/9261"
    ],
    [
      "1467619707276346/2401",
      "-44986651224423911229032/117649"
    ],
    [
      "3574650810363496/19321",
      "429438418929977252073628/2685619"
    ],
    [
      "388279859453299/1369",
      "6906559061067702727621/50653"
    ],
    [
      "8060861704918624/24649",
      "553063665933446434170016/3869893"
    ],
    [
      "-8683288988459/16",
      "-11541576049989069707/64"
    ],
    [
      "-662163627094766/5041",
      "-108698595972137713585246/357911"
    ],
    [
      "-503458984024",
      "235404335689709812"
    ],
    [
      "-445892920354",
      "-284040938896999428"
    ],
    [
      "263556983494989629/3157729",
      "1176719563136239432300438221/5611284433"
    ],
    [
      "-1468569612281/4",
      "-2543802760747785649/8"
    ],
    [
      "-1096876515952848974/5067001",
      "-3688999423907803208594319638/11405819251"
    ],
    [
      "-257456095898681/3844",
      "-66938026526589268162639/238328"
    ],
    [
      "30456837860889469/214369",
      "-17843128408934795320181261/99252847"
    ],
    [
      "-16165447381049/121",
      "-405138071564040335773/1331"
    ],
    [
      "1719231313751083594/13965169",
      "-9878449682385014825412626914/52187836553"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,14,37,83,84,98,139,148,157,180: 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 04:53:18",
  "updated_at": "2026-11-04 04:53:18"
}