{
  "id": 6490,
  "curve_key": "1350839493090506406889161:-7262613891785731042505391187436792325",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-28142489439385550143524",
    "8405803115499779884518863965587724"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 169.7516063040286,
  "faltings_height": 12.202627660642246,
  "conductor": "196347104202630277828224678520976443046562484852749066157604334",
  "bad_primes": [
    "2",
    "3",
    "7",
    "127",
    "14353421",
    "7708862556316319877716933",
    "110893095984415313988881719"
  ],
  "discriminant": "-29097565253392985165480954342019770950545551185885388245172550942258348",
  "regulator": "4752201857280451808.56271155172708068957694213552375844824990955177053453695541",
  "points": [
    [
      "5876731876445/16",
      "-13966044415520067583/64"
    ],
    [
      "23357925907135817",
      "-3569861903138058639964912"
    ],
    [
      "1144115221051060/1089",
      "-38344740227667867353494/35937"
    ],
    [
      "16079830850094269/10000",
      "-2029968700303856031546847/1000000"
    ],
    [
      "-2893958874012691/21904",
      "321211581193622610278705/3241792"
    ],
    [
      "-595391014632109/2500",
      "5000404592164329193789/125000"
    ],
    [
      "-2500184025650053/27556",
      "-462259312545501688618465/4574296"
    ],
    [
      "1276413502936235/3364",
      "44643730147036337744981/195112"
    ],
    [
      "29355083690408789/35344",
      "4963365687244094194119145/6644672"
    ],
    [
      "709583296672455548/2493241",
      "-602847485083792325462354902/3936827539"
    ],
    [
      "530065979974305/1849",
      "-12291032535386656956379/79507"
    ],
    [
      "43063377763235597/85849",
      "-8731883541376489633576445/25153757"
    ],
    [
      "483026826040835/2116",
      "-11466126083657369136913/97336"
    ],
    [
      "26796656668095338/332929",
      "-15679537647748114489064710/192100033"
    ],
    [
      "6495298574990903/20164",
      "-518288239772822646365671/2863288"
    ],
    [
      "4764362545969046/10609",
      "-321081253231758926958934/1092727"
    ],
    [
      "2388697855670964/1681",
      "-116101672443446815659542/68921"
    ],
    [
      "-141717295325001958/727609",
      "-50031999534095396981192590/620650477"
    ],
    [
      "222508141322204/289",
      "-3270512945174438893490/4913"
    ],
    [
      "362476875717",
      "-214080660612631105"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,99,100,148,150,166,174,188,197: 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 04:53:04",
  "updated_at": "2026-11-04 04:53:04"
}