{
  "id": 6120,
  "curve_key": "526011466494817165381441:-381699775775197790723225053255961825",
  "ainvs": [
    "1",
    "1",
    "1",
    "-10958572218642024278780",
    "441782147883394334171891116337501"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 163.85988488223452,
  "faltings_height": 11.405560686878738,
  "conductor": "1403350426911985538403637610899460834214448220801709234",
  "bad_primes": [
    "2",
    "3",
    "29",
    "41",
    "271",
    "152599",
    "383219",
    "12412664421137194517022733363095338101"
  ],
  "discriminant": "-88903418749089166432445734717963639955078242223752337359551725568",
  "regulator": "8575367206665900841099020.037089947601385478868181437442381646434732287161243",
  "points": [
    [
      "-28357747469523582417227/245060641296",
      "1536281762524663312437694839349739/121313839624606656"
    ],
    [
      "368220653948549/6241",
      "383559140002229199791/493039"
    ],
    [
      "18210115034715893/299209",
      "84301581995509879134467/163667323"
    ],
    [
      "22437090456217253/32761",
      "3323698417728949678732549/5929741"
    ],
    [
      "79273511405260587/1394761",
      "2624676334834030493439593/1647212741"
    ],
    [
      "502072268202487421/1874161",
      "331878003079611223556945569/2565726409"
    ],
    [
      "1078946195449844653/11744329",
      "583804233097548784252335899/40247815483"
    ],
    [
      "4962924100987418229/91145209",
      "2221852536783730060763524411/870163310323"
    ],
    [
      "17059582032256804885/57320041",
      "66586747924585486548845467987/433970030411"
    ],
    [
      "254273850687663745485/4206601",
      "4054635992199165491761548489947/8627738651"
    ],
    [
      "746273666216948427117/8484436321",
      "9835329848054721784673968637399/781509913963631"
    ],
    [
      "12231260456008587511569/209749344256",
      "98068244030567819235908907218311/96061843679739904"
    ],
    [
      "6833107484183966686541/110952279025",
      "25418674662879136587923200525151/36957649381832375"
    ],
    [
      "72401963782465642620229/1171791095049",
      "953560490394229628879854385592157/1268455657852877157"
    ],
    [
      "96362292514162262859709/1508045400625",
      "2895663557276311664005433655365377/1851917453102515625"
    ],
    [
      "147190555159309561059157/173458089289",
      "56059458798825927178177158136650403/72242345401350587"
    ],
    [
      "4039812493355488942176333/88206339263329",
      "4967276655003487997371361242228084343/828418325839136358767"
    ],
    [
      "235165257867372865142658765/8174916890071009",
      "9062950671428679204738034216407578432951/739137129820242126274223"
    ],
    [
      "3323381603472582456595770421/60480721363041225",
      "34981255826156917588550134433120183369789/14873920080375590848549125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,5,23,24,45,76,86,106,114: 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 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 03:13:59",
  "updated_at": "2026-10-29 03:13:59"
}