{
  "id": 6517,
  "curve_key": "738459091136020176:-658179270713082295905036480",
  "ainvs": [
    "0",
    "0",
    "0",
    "-15384564398667087",
    "761781563325326731371570"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 123.5030391479349,
  "faltings_height": 8.214833024835974,
  "conductor": "6810058810005453132518759329541962443430063224",
  "bad_primes": [
    "2",
    "3",
    "1721308271520361",
    "54948989500539799425066454847"
  ],
  "discriminant": "-17651672435534134519488624182172766653370723876608",
  "regulator": "155866337221101.604160058433357152740957920067288092827540300692270609670",
  "points": [
    [
      "-572475543/4",
      "1433249256573/8"
    ],
    [
      "441718858437657/256",
      "-9283653242106855044733/4096"
    ],
    [
      "-945021017913/6889",
      "-308344365203381928/571787"
    ],
    [
      "11649879946101/11449",
      "-39481239778078976838/1225043"
    ],
    [
      "6233082346417/15129",
      "-14928251088894178598/1860867"
    ],
    [
      "5697977627721/85264",
      "-4461760918817580357/24897088"
    ],
    [
      "2218610620854/22801",
      "1485156041809957656/3442951"
    ],
    [
      "60504065433/484",
      "9476723282597829/10648"
    ],
    [
      "5654586825747/26569",
      "11561989999879805832/4330747"
    ],
    [
      "-11014675703399/83521",
      "17017060699180670498/24137569"
    ],
    [
      "-1493616820703/11881",
      "1090463417468081558/1295029"
    ],
    [
      "200396397033/3844",
      "-75904043026151619/238328"
    ],
    [
      "-320085924671/2500",
      "99427483812147767/125000"
    ],
    [
      "830119369113/33856",
      "-3936465662253039549/6229504"
    ],
    [
      "20247285",
      "677190024630"
    ],
    [
      "-3365429943072663/35081929",
      "241860517208570847206898/207790265467"
    ],
    [
      "21838554764833/401956",
      "-74864919672557906599/254840104"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,8,83,107,123,146,151,154,163,183: 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 >= 17: the 17 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 04:56:15",
  "updated_at": "2026-11-04 04:56:15"
}