{
  "id": 6529,
  "curve_key": "5866960774607296:-455581990090963571724800",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-122228349470985",
    "527294010755472476121"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 108.95172530715317,
  "faltings_height": 6.957803726700435,
  "conductor": "1826808848768856518007197023931144064",
  "bad_primes": [
    "2",
    "3",
    "11",
    "29",
    "503",
    "29648533316312556317125931453"
  ],
  "discriminant": "-3244763262712452796432239296330306637324288",
  "regulator": "401989327481.93567156628280654861707814682180850238022777544453261494",
  "points": [
    [
      "58028089",
      "-434545998252"
    ],
    [
      "1178452341",
      "-40452837723924"
    ],
    [
      "-3900211933/529",
      "-390044319862684/12167"
    ],
    [
      "1099609879/289",
      "53213389387632/4913"
    ],
    [
      "1997538765/361",
      "30965702109324/6859"
    ],
    [
      "1726083279/361",
      "-49547915947968/6859"
    ],
    [
      "52657282896/9025",
      "-3063254980629981/857375"
    ],
    [
      "177690381/25",
      "524490365604/125"
    ],
    [
      "339423725817/37249",
      "93773033408705556/7189057"
    ],
    [
      "3468495428181/450241",
      "-1978119141307473036/302111711"
    ],
    [
      "229280859/25",
      "-1666375797252/125"
    ],
    [
      "-964793521/169",
      "-70817732586128/2197"
    ],
    [
      "7255845",
      "-4735456836"
    ],
    [
      "7496691",
      "-5683541124"
    ],
    [
      "1990882266891/265225",
      "-781730974411553364/136590875"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,23,34,57,76,95,105,193,198: 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 >= 15: the 15 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 04:57:39",
  "updated_at": "2026-11-04 04:57:39"
}