{
  "id": 272,
  "curve_key": "0:-1028842570723968",
  "ainvs": [
    "0",
    "0",
    "0",
    "0",
    "1190790012412"
  ],
  "rank_lower_bound": 6,
  "naive_height": 69.13442169512868,
  "faltings_height": 3.8188577170523517,
  "conductor": "35280726487742736",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "19",
    "823"
  ],
  "discriminant": "-612567728781193921272945408",
  "regulator": "1643.7083603849346516610625128373530480382179836418",
  "points": [
    [
      "-10582",
      "76362"
    ],
    [
      "-8883",
      "699895"
    ],
    [
      "-6006",
      "986986"
    ],
    [
      "-5187",
      "1025297"
    ],
    [
      "-2926",
      "1079694"
    ],
    [
      "218",
      "1091238"
    ]
  ],
  "submitter": "sorinmg",
  "commentary": "This is the Mordell curve\r\n\r\nE : y² = x³ + 1190790012412,\r\n\r\nwith 1190790012412 = 38038²·823 = 2²·7²·11²·13²·19²·823. It has j(E)=0 and geometric complex multiplication by Z[ζ₃], while End_Q(E)=Z; equivalently, E is a sextic twist of y²=x³+1.\r\n\r\nThe minimal discriminant is\r\nΔ = -612567728781193921272945408\r\n= -2⁸·3³·7⁴·11⁴·13⁴·19⁴·823²,\r\n\r\nand the bad primes are 2,3,7,11,13,19,823. Their Kodaira types are respectively I₀*, III, IV, IV, IV, IV, II. The conductor is\r\n\r\nN = 35280726487742736\r\n= 2⁴·3²·7²·11²·13²·19²·823²,\r\n\r\nand the Tamagawa product is 324.\r\n\r\nRational torsion is trivial: #E(F₅)=6 and #E(F₃₁)=43, whose gcd is 1. Six independent rational integral points give rank(E/Q) ≥ 6, while the supplied PARI/GP computation gives the matching upper bound, hence rank(E/Q)=6 and E(Q) ≅ Z⁶.\r\n\r\nThe submitted six-point lattice agrees with the PARI generator lattice via an integral transition matrix of determinant 1. It has been saturated at ℓ=2,3,5,7,11,13. Therefore, if its index i in the full Mordell-Weil group is greater than 1, every prime divisor of i is at least 17. The displayed height determinant 1643.70836038493465... is the regulator of this rank-6 lattice; the global regulator is det(H)/i².\r\n\r\nA rational 3-isogeny exists to\r\n\r\nE' : Y² = X³ - 32151330335124,\r\n\r\nwith\r\nX = (x³+4B)/x²,\r\nY = y(x³-8B)/x³.\r\n\r\nIts geometric kernel is {O,(0,±38038√823)}, a Galois-stable subgroup defined over Q(√823). Consequently E' also has rank 6 over Q.\r\n\r\nAn exhaustive integral search for -10599 ≤ x ≤ 10¹¹ finds exactly 30 points with y>0 (60 after sign symmetry). Independently, searching combinations of the Mordell-Weil generators with coefficients -5≤n_i≤5 produces exactly the same 30 positive integral points.\r\n\r\nΩE​≈0.0408588818808802051405413396566268225117823…\r\nso... \" i think\" it's a real testing ground / candidate for BSD\r\n",
  "created_at": "2026-08-18 22:06:22",
  "updated_at": "2026-08-18 22:06:22"
}