{
  "id": 6487,
  "curve_key": "40420360623634734070830481:-334558559142321224415624153918345480121",
  "ainvs": [
    "1",
    "0",
    "0",
    "-842090846325723626475635",
    "387220554562801613280889701177360225"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 177.4117505534648,
  "faltings_height": 12.79746219564097,
  "conductor": "4696433430798031027581017502847138013984755935928291846619310",
  "bad_primes": [
    "2",
    "3",
    "5",
    "101",
    "51421",
    "3488029",
    "88258214593",
    "898391459550157",
    "108989380645966881553"
  ],
  "discriminant": "-26556955935454327797485778256675731827073300081136450588779507916032000000",
  "regulator": "1121320570323967637.42088114845475370533542412555879763706621262352410808886213",
  "points": [
    [
      "125778359089204/25",
      "-1389133340141236081177/125"
    ],
    [
      "11251687731070",
      "-37621559406100118135"
    ],
    [
      "-35795849897870/169",
      "1638321732037423341505/2197"
    ],
    [
      "827440287600250/2809",
      "60424152446643780015905/148877"
    ],
    [
      "225433650519670/361",
      "2221298789429895706435/6859"
    ],
    [
      "110122661109400/169",
      "745614381371793555955/2197"
    ],
    [
      "4983521860634530/6889",
      "226282112782160431516655/571787"
    ],
    [
      "6473381944682170/7921",
      "348836853889191393137885/704969"
    ],
    [
      "18475250636787970/14161",
      "2070261395838547833820235/1685159"
    ],
    [
      "744484747950",
      "-415852567137910215"
    ],
    [
      "294446353829314966/358801",
      "-107208547897369695133101817/214921799"
    ],
    [
      "96297457021750420/120409",
      "-19831328484254630151844405/41781923"
    ],
    [
      "1224491690157280/1369",
      "-29949488739705918925805/50653"
    ],
    [
      "366463083689350/289",
      "-5725962645001521408895/4913"
    ],
    [
      "675713504725630/1369",
      "-15349435541547311067755/50653"
    ],
    [
      "3682180717404828070/4330561",
      "-4818942612358659806995433335/9011897441"
    ],
    [
      "13696509363166/25",
      "-37565223806477341579/125"
    ],
    [
      "-534584245095770/8649",
      "-532963453307239473788795/804357"
    ],
    [
      "2262327724456650/3481",
      "-69479857179857547755565/205379"
    ],
    [
      "-21410873090",
      "-636585128544467975"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,10,39,53,76,78,83,89,119,165: 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 04:52:42",
  "updated_at": "2026-11-04 04:52:42"
}