{
  "id": 6509,
  "curve_key": "1221310897388110692340641:-1278729779784720489095784581980840689",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-25443977028918972757097",
    "1480011319201639338118091527889321"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 166.38590105936896,
  "faltings_height": 11.807429913076367,
  "conductor": "62209683646987441959784598983560125347538651778630",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "271",
    "10037",
    "288348405564539",
    "125900273648686732933892417"
  ],
  "discriminant": "107961725171300801101318183536361157187915956416016410364312640000000",
  "regulator": "684494388065104.93579919480830319110777074028959403483343132598067696054248",
  "points": [
    [
      "10431942550141/81",
      "13427996526674870626/729"
    ],
    [
      "329154499754871",
      "-5971723015977112106276"
    ],
    [
      "5695314695569/49",
      "3305451863647336402/343"
    ],
    [
      "1289329516973449/1849",
      "-45172398882623002443702/79507"
    ],
    [
      "20766012664929/289",
      "-23425803671593990978/4913"
    ],
    [
      "-2127280339604609/11881",
      "-22268531253176086377524/1295029"
    ],
    [
      "10001753106254511/22201",
      "944080664453086258706156/3307949"
    ],
    [
      "20303602246356639/24649",
      "2842210733359985768682892/3869893"
    ],
    [
      "8016668581786089/6889",
      "711343860202761785444618/571787"
    ],
    [
      "581459911179/4",
      "233655868187055737/8"
    ],
    [
      "-34107763887332121/187489",
      "-762637070036227446691172/81182737"
    ],
    [
      "3748190260983441/33856",
      "27890741587708410415651/6229504"
    ],
    [
      "282811193133651/6241",
      "10105107807708409992016/493039"
    ],
    [
      "3247174160073711/22201",
      "98544712055249889016556/3307949"
    ],
    [
      "565953812803550031/2886601",
      "311265677242706293374916996/4904335099"
    ],
    [
      "-862135795991361/11449",
      "-66750794839895353354108/1225043"
    ],
    [
      "250710888894448479/1841449",
      "58044271622093726055579532/2498846293"
    ],
    [
      "1246828240542249/9409",
      "-19041497707729703540288/912673"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,18,70,86,102,109,149,157,166,195: 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 >= 18: the 18 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 04:55:19",
  "updated_at": "2026-11-04 04:55:19"
}