{
  "id": 4967,
  "curve_key": "224915288951058674136645381085801494337761:-150983519505859939262473089994054028263141651882502009189517009",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-4685735186480389044513445439287531132037",
    "174749443872523077851256028697145592936060828816421430456061"
  ],
  "rank_lower_bound": 18,
  "torsion": [
    2
  ],
  "family": null,
  "families": [],
  "naive_height": 286.3445525363118,
  "faltings_height": 21.8859430918817,
  "conductor": "483328727128257581098535465545997674973027785591045803016636754582035590",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "37",
    "41",
    "43",
    "71",
    "113",
    "127",
    "131",
    "149",
    "337",
    "401",
    "1039",
    "1579",
    "16661",
    "22961",
    "32183",
    "131581",
    "162947",
    "164117",
    "1974221",
    "2835163"
  ],
  "discriminant": "-6607788663060192289559555973239565730544109081541180627287355929604811916203253276349776833194685554207449466121917488000",
  "regulator": "88035820919908594011786240.260236482845755954358994494011133510084030172422",
  "points": [
    [
      "7863158345912610472801/841",
      "8852777941696951833932176774840036/24389"
    ],
    [
      "21965081164231135561",
      "287096270377070130604930634324"
    ],
    [
      "40254108965326834467019/1369",
      "12652364428717057709230688868869612/50653"
    ],
    [
      "63108011304078971245",
      "361077985411308288516733872062"
    ],
    [
      "84137685282135336391",
      "613290985806479800397868632264"
    ],
    [
      "-23019882189146848109",
      "520015322398270426452251949344"
    ],
    [
      "43983340871304709771",
      "231824149587817279164518990324"
    ],
    [
      "66182304335781266791",
      "393092395399938145086557964464"
    ],
    [
      "131005012953927277411",
      "1345081312291533306068492584124"
    ],
    [
      "-36224319655386246484084907669/446772769",
      "1389572358258918230413619460275278775800192/9443436018353"
    ],
    [
      "-21674719256391391691819/361",
      "3357716636149455907602496120194716/6859"
    ],
    [
      "812413641963566883173221/14161",
      "518718640391255572064590722463110316/1685159"
    ],
    [
      "-68953814302970692897199/841",
      "2148785530195679287830679437650596/24389"
    ],
    [
      "8417178434171609057665091341/105042001",
      "603078462101057116424188362765757045453886/1076575468249"
    ],
    [
      "4366334491187244745502563091/6630625",
      "287045408788598390579699495391415296494164/17073859375"
    ],
    [
      "33404353866071907582128270039/16785409",
      "6101721291402947842765004895200094592672592/68769820673"
    ],
    [
      "-670175859795981349864135588519/9232903744",
      "322858206306729819776841791967855903167490233/887171254953472"
    ],
    [
      "31254435622693548906757304941111/867433312321",
      "185454074152488366173553430648976912920388731384/807893557196598881"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 2-isogeny from curve #4578, submitted by jesper-petersen: https://elliptic-rank.icarm.cloud/curve/4578.\r\n\r\nThe source commentary attributes the curve and independent-point witnesses to the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r18-z2k3.ec: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/2-r18-z2k3.ec.\r\n\r\nIt identifies the specialization u = 4/7, t = -12673/12022 of the Elkies–Klagsbrun Z/2Z family, citing https://arxiv.org/abs/2003.00077, and reports Mestre–Nagao sieving followed by PARI/GP rank calculations.\r\n\r\nApplied a rational degree-2 isogeny and converted the target to its minimal model y^2 + x*y + y = x^3 - x^2 - 4685735186480389044513445439287531132037*x + 174749443872523077851256028697145592936060828816421430456061.\r\n\r\nEighteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 483328727128257581098535465545997674973027785591045803016636754582035590. Source-attributed existing curve.",
  "created_at": "2026-10-21 00:03:24",
  "updated_at": "2026-10-21 00:03:24"
}