{
  "id": 5856,
  "curve_key": "40543658826670537741997684379216:-240808901279849764195747286987721006268270726080",
  "ainvs": [
    "0",
    "0",
    "0",
    "-844659558888969536291618424567",
    "278714006110937227078411211791343757254942970"
  ],
  "rank_lower_bound": 23,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 218.3397965371019,
  "faltings_height": 16.148983589744933,
  "conductor": "5819267853110420154890366735286544171416196785322380671321316665885175203177478347976",
  "bad_primes": [
    "2",
    "3",
    "7",
    "61",
    "5107",
    "5153",
    "1758673073707969",
    "64916681682324402712972707812433037359687444023609926367"
  ],
  "discriminant": "5009411984528749202774584177470186392168547511102635598054871668127267140501675041244876032",
  "regulator": "1571639839654745583155087.45033233975303515074227519351361646447496406340417375178988",
  "points": [
    [
      "400140502141191",
      "-2190655335354102165612"
    ],
    [
      "-311770413262539",
      "-22621881778523540047542"
    ],
    [
      "641321014505451",
      "-886959060166350263652"
    ],
    [
      "-562270685737362",
      "-23997507798719700488964"
    ],
    [
      "355604786701234",
      "4828762091069806386014"
    ],
    [
      "365997995144985",
      "-4312469280454662677910"
    ],
    [
      "649525552624809",
      "2027342135532900957414"
    ],
    [
      "5780767988836813/9",
      "29236764845375278503326/27"
    ],
    [
      "83522716404285",
      "-14448128026609993441650"
    ],
    [
      "1594205000329585",
      "-54624252080729087854930"
    ],
    [
      "11785465140650313/16",
      "-479766465813352206526779/64"
    ],
    [
      "14861678231228859/49",
      "-2435779463958109866906084/343"
    ],
    [
      "287859018827037",
      "7708692677026688070738"
    ],
    [
      "-586331039182623",
      "23924732845870104100062"
    ],
    [
      "411881037643719",
      "829889505711096146484"
    ],
    [
      "-613101302695413",
      "-23793176897777808981012"
    ],
    [
      "-1052932595101653",
      "-855519462521213670012"
    ],
    [
      "407459056885533",
      "1482344831942816443986"
    ],
    [
      "-1027842507788943",
      "-7811291993171481908562"
    ],
    [
      "32129627228098393/36",
      "3316608116390412417853349/216"
    ],
    [
      "7264060315633341031",
      "-19578037630902093994283960572"
    ],
    [
      "838319419295170113/1024",
      "-382007375331011799135378591/32768"
    ],
    [
      "1645267718747989",
      "-57815289847419623060374"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -79/21 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -129077164062048809835900T^8 + 378319182388641891290880T^7 + 136088129918041916509980T^6 - 755961738076060064377560T^5 + 619450622573441931032760T^4 - 1170861959203596215660160T^3 - 1734751811488996657912320T^2 + 1314050376486783211611840T - 219065110811691252584112,\nA6 = 18245050345346515787515057919752500T^12 - 83505322013254826025359146692690000T^11 - 2407103439528010819936132599838344T^10 + 465220783429849849086641780587731600T^9 - 185239629309258027189063745343474940T^8 - 1503733795718269687786747306505656800T^7 + 626403979186422524394828379594167600T^6 + 2519798602559624814688441138399867488T^5 - 832465009702733039267308016691100320T^4 - 905573011434584900869203229631278080T^3 + 930446348033589270658482617988215040T^2 - 337397396218794978267255003741415680T + 42647307174629128234916120026233216.\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": "2026-10-25 13:12:12",
  "updated_at": "2026-10-25 13:12:12"
}