{
  "id": 6167,
  "curve_key": "1006455639872459634296049:-907119077073355831806882605821973193",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-20967825830676242381168",
    "1049906339210514909937857261778131"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 165.80543137000413,
  "faltings_height": 11.791740757618772,
  "conductor": "19790611384047910958418807384150866111612560217535236970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "29",
    "5494134725747",
    "41438540727817",
    "1585972493424107581091963"
  ],
  "discriminant": "113788885721116751303535254279442348583924098910605730046279929036800",
  "regulator": "729226925067455932207458.8192822595874263602839180195139509331520834643669569",
  "points": [
    [
      "-140131485370269694147/2347305601",
      "5197725320303107975530919499409/113724609062849"
    ],
    [
      "-45664090007519431987/325477681",
      "205942819228572118369061730201/5871942842921"
    ],
    [
      "-22047857600262907/12866569",
      "1520811255726888563496581163/46152383003"
    ],
    [
      "-433097155402891/2809",
      "3699660033828541214589/148877"
    ],
    [
      "41347617187173/361",
      "84251356854542136539/6859"
    ],
    [
      "173760483563093/3481",
      "2320288704401108058195/205379"
    ],
    [
      "535723345209605/841",
      "12100900775974373009013/24389"
    ],
    [
      "1369357183846421/24025",
      "23541423931284047336691/3723875"
    ],
    [
      "66939723957464309/1113025",
      "2969427094463251585870263/1174241375"
    ],
    [
      "359377060396551365/6538249",
      "133184046083019469983677781/16718302693"
    ],
    [
      "709757678181763925/2343961",
      "537866576691451061787147027/3588604291"
    ],
    [
      "812225328739909385/102839881",
      "31021575425998632778292309037/1042899233221"
    ],
    [
      "5601515185716686867/107474689",
      "11066676113593233598851770733/1114190100863"
    ],
    [
      "5958294093388343525/52519009",
      "4361432270073030286779554871/380605258223"
    ],
    [
      "16098699469336757525/20620681",
      "63544774599369039862860694437/93638512421"
    ],
    [
      "206503443823409936429/3400405969",
      "142972947303542357070817123017/198287873270297"
    ],
    [
      "1333211851953757651421/10322356801",
      "23364077853643884344653006912623/1048741128624799"
    ],
    [
      "7544264507825807360445/47429435089",
      "430764929056768961414580552858599/10329324661987687"
    ],
    [
      "5735832129723243720243085/26329305801681",
      "11157815443576468271724982057893901193/135101170893337762329"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,30,34,43,48,71,84,96,121: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:39:09",
  "updated_at": "2026-10-30 14:39:09"
}