{
  "id": 6170,
  "curve_key": "63256181978664071369361:-15144040906988534629373950417874745",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-1317837124555501486862",
    "17527825124158781768840206873885"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 157.5044394434987,
  "faltings_height": 11.062969486944024,
  "conductor": "174117693036610424661072980305410965823814139317254",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "139",
    "227",
    "281",
    "2267",
    "6250016683150324800955592393624539069"
  ],
  "discriminant": "13754516528009961126928258706015332744857844843096473684280983552",
  "regulator": "3512130842499827869410.3526012722933445876108697295973084380057956384840232",
  "points": [
    [
      "-20841392259040239431/1388829289",
      "301460726601944158676471158073/51757501113163"
    ],
    [
      "-2911952346908634623/100620961",
      "5658409360746517134270420757/1009328859791"
    ],
    [
      "-430256846813219855/12538681",
      "209875721769642908164374839/44399469421"
    ],
    [
      "-198491323162019/4761",
      "22319221690233865139/328509"
    ],
    [
      "32989816717/1369",
      "211872895993371842627/50653"
    ],
    [
      "3808719957373/121",
      "3579791609169655837/1331"
    ],
    [
      "7546750581665/169",
      "15177785908165319503/2197"
    ],
    [
      "15379386654497/961",
      "21820404224928407197/29791"
    ],
    [
      "799543110374437/29584",
      "6540221611990968923657/5088448"
    ],
    [
      "439400878619545/3481",
      "8863348279870931373961/205379"
    ],
    [
      "850301936235823/1681",
      "24732481534134793637989/68921"
    ],
    [
      "6041119353683749/4225",
      "469393671833479140289963/274625"
    ],
    [
      "99470876978072065/5822569",
      "211950714954298778513903/14049858997"
    ],
    [
      "351720525761647681/52548001",
      "1143203484548324900354989523/380920459249"
    ],
    [
      "13418585900892702889/14799409",
      "49115282128428368994685895861/56933326423"
    ],
    [
      "21542859625599730969/722480641",
      "42296939749245462325255123885/19419557149439"
    ],
    [
      "80980578438616118273/796876441",
      "687138120660376936111241601943/22495025052989"
    ],
    [
      "2250969825953868407413/9318427024",
      "105650071282939021657033623368141/899526397480768"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,65,67,79,88,98,118,129,139: 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 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-30 14:39:33",
  "updated_at": "2026-10-30 14:39:33"
}