{
  "id": 6486,
  "curve_key": "107853437825161540571241:-35895352430789533801885415573435445",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-2246946621357532095234",
    "41545546795457030407521576974544"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 159.1318317943297,
  "faltings_height": 11.138283593447394,
  "conductor": "2492992669762339391373056866125452214880825163390514171138798",
  "bad_primes": [
    "2",
    "3",
    "19",
    "23",
    "467418881",
    "41235613974633777513431",
    "5481091952843413380510791"
  ],
  "discriminant": "-19609880340350561652540465308942807122252570735229784470177785068",
  "regulator": "2103702594994226935.73476742595392354959001891784813913535252583245498714350503",
  "points": [
    [
      "533051318728/9",
      "-291090331253911624/27"
    ],
    [
      "15317673335478",
      "-59949723413749857780"
    ],
    [
      "-5433186562628823/153664",
      "527849597227259830085151/60236288"
    ],
    [
      "38469901900497/4096",
      "1209007313875919261817/262144"
    ],
    [
      "181943398799823/7396",
      "684304184080735056237/636056"
    ],
    [
      "29278279749097/1024",
      "26919073795094178811/32768"
    ],
    [
      "327845545096461/10000",
      "1765805696267956031241/1000000"
    ],
    [
      "7001074074049546/82369",
      "509553508661859248512234/23639903"
    ],
    [
      "398772279729997/2704",
      "7594891053397509069291/140608"
    ],
    [
      "-2295023571090/49",
      "-2276186841414390414/343"
    ],
    [
      "39685439322",
      "-3857017953083952"
    ],
    [
      "25458081546",
      "-917799961267296"
    ],
    [
      "213649800308084655/4553956",
      "-60994002406829837765611263/9718142104"
    ],
    [
      "398476877943/121",
      "-7781700649676884413/1331"
    ],
    [
      "-177071009456942/5929",
      "-4134412260787016144280/456533"
    ],
    [
      "404891128690250941/2839225",
      "-244936283766915862788552266/4784094125"
    ],
    [
      "8916313174322049997/184226329",
      "-16990056044680801424598076666/2500503963517"
    ],
    [
      "621487116833746/11025",
      "-11224242753302190317096/1157625"
    ],
    [
      "7761678087762/49",
      "-20751074404917855270/343"
    ],
    [
      "907687748860665135/11978521",
      "-725677702627828302240462243/41457661181"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,56,64,86,96,100,123,156,180: 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 >= 20: the 20 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 04:52:34",
  "updated_at": "2026-11-04 04:52:34"
}