{
  "id": 4264,
  "curve_key": "40142576247322913301546823595585116748576521:-254349805250732808707457202824277918970695812029610473376892228069",
  "ainvs": [
    "1",
    "0",
    "1",
    "-836303671819227360448892158241356598928678",
    "294386348669829639707635403888781174306416078465529045951121948"
  ],
  "rank_lower_bound": 25,
  "torsion": [],
  "naive_height": 301.203142728438,
  "faltings_height": 22.7770988476176,
  "conductor": "310185621680651834905251713151662581225990024407184633137813567866714832202772684700109205190",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "29",
    "41",
    "53",
    "59",
    "83",
    "89",
    "2011161233376865227863209973974465563186794920446510866861224845742039487039"
  ],
  "discriminant": "-4059714867846464612952509579831371227530801741091951999101234159870332617575772149629696350028177466955544688612536096312500",
  "regulator": "18421311845773731515083879.98028731576732728664879916606541034291584544626532793492962783",
  "points": [
    [
      "511379322488021094824",
      "669434398307403899721512850075"
    ],
    [
      "726129004915446929084",
      "8365595858752351292629657912335"
    ],
    [
      "55120435932686342738924/25",
      "-11969095558138296059029440934962492/125"
    ],
    [
      "523265536976173710344",
      "226125185414371531248786233595"
    ],
    [
      "-851170966621133202601",
      "19737203074288440094091088521175"
    ],
    [
      "545905486034012307647",
      "-728291485944177624427091371233"
    ],
    [
      "2003358660892190131031/4",
      "-8627786617422603193976740937115/8"
    ],
    [
      "245315004920047561124",
      "10197618926750918747107893657975"
    ],
    [
      "773181364976835731399",
      "10487470616623709676103529917425"
    ],
    [
      "1066303509719447696471/4",
      "-76059582852469742553886142162475/8"
    ],
    [
      "80057016241196572360411/9",
      "22536348190332123285757767056169305/27"
    ],
    [
      "776086027071977520524",
      "-10620111022669795638854486510700"
    ],
    [
      "-1055760583765885752976",
      "-734812114048395815203503695700"
    ],
    [
      "508714839085709323784",
      "772641350245262457622466743035"
    ],
    [
      "571444960249395515909",
      "1757784477120389421596258714535"
    ],
    [
      "509945675928699735760559/961",
      "4909043627806196370461861007842385/29791"
    ],
    [
      "523920217465390457099",
      "-205090129708185143328605281725"
    ],
    [
      "987414742112950399874",
      "20768369539665126985790734763400"
    ],
    [
      "6375603367935464596949",
      "504103164963805638291255047348475"
    ],
    [
      "425303614448117053274",
      "3953939542528384495515327511800"
    ],
    [
      "62969704858987737146279/121",
      "-434139571386113181806238383153325/1331"
    ],
    [
      "61900725605932030435379/121",
      "880882533906642422807078888542275/1331"
    ],
    [
      "542365797497305357019",
      "-588660158689383809729830845165"
    ],
    [
      "536586432781071528299",
      "365777612229931012882180935075"
    ],
    [
      "2309063290191667923539",
      "103318489093150477131199905849345"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -4927/4396 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -234114040453021356506112T^8 - 272002889947768518314880T^7 + 686028086039421713758320T^6 + 1142195364874563315587280T^5 + 627637229082985727538360T^4 - 984449918457333810927360T^3 - 2633542817168016219118320T^2 - 1006973197204579283167560T - 114715927313180034201675,\nA6 = 48308501179046578125484559991177216T^12 + 57069826702680132557515086694133760T^11 - 311131671782769127044180046463351040T^10 - 240521895686500160053739383108517760T^9 + 1147866319247799475598436207881897040T^8 + 683399028950475828317237644205746752T^7 - 2372093631572614744214600133762992400T^6 - 1765634230326260432804074074782197200T^5 + 1564909970672513896400523764894111640T^4 + 1894740665595018649008096627125850240T^3 + 882898465151004944410908107862516576T^2 + 191456284444900580427321522994041240T + 15250200281398537130338745700198150.\nIt is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.",
  "created_at": "2026-10-15 21:54:58",
  "updated_at": "2026-10-15 21:54:58"
}