{
  "id": 4268,
  "curve_key": "1366346104772775183581927130797148936075130931041:-1652307340108211323372750601401018454904483109576945432607675336760883345",
  "ainvs": [
    "1",
    "0",
    "0",
    "-28465543849432816324623481891607269501565227730",
    "1912392754754874216866607120233635758960309880423899579727165768593156"
  ],
  "rank_lower_bound": 28,
  "torsion": [],
  "naive_height": 332.57659878918855,
  "faltings_height": 25.63428206002427,
  "conductor": "13254623199995407422640596558711966162848609362761245598355101472245794255676684213067251702177179460045793902",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "17",
    "19",
    "23",
    "29",
    "41",
    "2633",
    "1228033363",
    "413206565930665426383996259338260620026299091043801666189715739305939401348860205909"
  ],
  "discriminant": "-103753304634559133523953656302161172553189908351064859811767624372452282326727675614470347349863123677378504676170305794066145985763852320768",
  "regulator": "6043890484070334802354174130283.163081313470506468746506527844937793372855914284051265581974701",
  "points": [
    [
      "618909920991125770930714",
      "470497518630730070220324426734957170"
    ],
    [
      "127034177755142142970660",
      "-18610024224135281799610001406957314"
    ],
    [
      "110184867717260110881988",
      "-10660326156599217108354165359377826"
    ],
    [
      "78311794873294048023748",
      "-12785561495500595424791176707405602"
    ],
    [
      "-194121207168249218839100",
      "11094173178336252704911115923338334"
    ],
    [
      "427965943479555029815531636/3721",
      "2870064512132323961256279882468807384838/226981"
    ],
    [
      "4319289986551188608786404/49",
      "-3219893586501545737786441887032992142/343"
    ],
    [
      "75310294173376220901356575/4",
      "653527910670700903498583205489682790153/8"
    ],
    [
      "-138637666599806384062262",
      "56516524706819593710556148509441924"
    ],
    [
      "150910152076489555803844",
      "-32457079230498361077230655818333474"
    ],
    [
      "1893153182955188933264164/25",
      "1727782050425516821670598114564325478/125"
    ],
    [
      "434049471507710909376316",
      "-267079389776596462944820050248446202"
    ],
    [
      "67760781422403093379236",
      "17165968662219866687168130912444798"
    ],
    [
      "5521284668883536360696644/25",
      "9998311030857444346691735975082153062/125"
    ],
    [
      "3349064478944747229480604",
      "-6121312489230166403533247610035269850"
    ],
    [
      "-181404426862380353983484",
      "33265451401704296782988773291310686"
    ],
    [
      "6407142507915677833831669486/49729",
      "217205031761376199057957880282379677566720/11089567"
    ],
    [
      "98327696107367231434720",
      "8006659487223652453355971076764606"
    ],
    [
      "1494741440805808218471082564/12769",
      "19588183926034433094869840847648825990206/1442897"
    ],
    [
      "18685932961045851000573292",
      "80770828873825810242984897584665403134"
    ],
    [
      "46270681818537301078744",
      "-26350276119942308456489658737424674"
    ],
    [
      "2972093877249190705842586/25",
      "1805059734419236402303057916135595626/125"
    ],
    [
      "76152370182268197787798825108/11449",
      "21008100669727862510534989660878192465512762/1225043"
    ],
    [
      "-1996989879531652790696156/25",
      "-7579286296551167127450501738979014682/125"
    ],
    [
      "419987770012219683994280932/3721",
      "-2660675364909855981019166888796464383466/226981"
    ],
    [
      "-173669409687195941272964",
      "-40223535339512068604985845184571202"
    ],
    [
      "66066329997207421342328344540/97969",
      "-16495814646571287090564858066257422504538258/30664297"
    ],
    [
      "-29981488566468615905708",
      "52334329863415820533397784530043934"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = 3904/4935 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -11696112421636110725232T^8 - 225953091010175473234080T^7 - 1285862935447903797580320T^6 - 1129158151626029959937952T^5 - 1592088393814170815994480T^4 - 37631263114247380337808T^3 + 1695532195366906279824008T^2 + 289018889748535735084344T - 624976459672410496278747,\nA6 = 449725080902042019431313876378240T^12 + 11412860638353264300064607972424576T^11 + 108724211594005774585272702173979840T^10 + 552699339212501471676010717898982720T^9 + 1706301350149364859650702774642594064T^8 + 698629245250434441649386767885626848T^7 - 1435130050396849573556196386817943392T^6 - 607341322268570723710732507501947360T^5 + 1272148652815891722874496655477939744T^4 + 339936587466097075261357997400067920T^3 - 735579972865799521796462939416343048T^2 - 132384785109517617529019092091956392T + 190126813773008796183739475605546614.\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": "2026-10-16 00:28:16",
  "updated_at": "2026-10-16 00:28:16"
}