{
  "id": 5330,
  "curve_key": "1705238466897235334533112006502321:-91169136941944424863945313802294816063036671547369",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-35525801393692402802773166802132",
    "105519834423546796918424091249830811877361922231"
  ],
  "rank_lower_bound": 23,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 230.0736017855927,
  "faltings_height": 17.185084747167046,
  "conductor": "118911634774286608520500346082302161023536817904543003128504505533545853783719245670",
  "bad_primes": [
    "2",
    "5",
    "13",
    "23",
    "41",
    "13093667",
    "5065697023",
    "1273551055088309",
    "17180854887289923367",
    "51411978794385181442027687"
  ],
  "discriminant": "-1940540467105150355464865654179660645973810386840914409429030659416535252986878466528347136000000",
  "regulator": "268378835834197466063012.546073532080951070557232094052514265833979694114498015938689",
  "points": [
    [
      "19472919573827471449/1089",
      "81852866517314407020961127873/35937"
    ],
    [
      "-9680915988411038275/1849",
      "30586505953667061558492621079/79507"
    ],
    [
      "2627819121719097349/289",
      "-3591112292723399375084831523/4913"
    ],
    [
      "4972523169902912589/169",
      "-10881841930257278866163668047/2197"
    ],
    [
      "13869748172053381",
      "-1510267593649995474278451"
    ],
    [
      "675796135691736809/49",
      "-513231912393134735662307753/343"
    ],
    [
      "-174933683111300091/49",
      "148264681149397156067769947/343"
    ],
    [
      "-47451931761053793119/20736",
      "-1248530309774951675679309393719/2985984"
    ],
    [
      "22540182353585309",
      "3279713816809025377259141"
    ],
    [
      "31672553325400489141/1521",
      "-171873709967724961096474986229/59319"
    ],
    [
      "166753042862093779129/6561",
      "2100386384437228144896695786549/531441"
    ],
    [
      "-11520087584872810531/2809",
      "-63554558956480159518079965367/148877"
    ],
    [
      "-2203941579133301081/484",
      "4427187325459845009115601017/10648"
    ],
    [
      "-7029991569323089",
      "-88533841245488048378851"
    ],
    [
      "-73701663823368099/121",
      "-474203718754523023841512361/1331"
    ],
    [
      "-3456897372619199259/961",
      "12874609640736672558876067059/29791"
    ],
    [
      "419577430762894042269/41209",
      "-7479045446339265985478219570217/8365427"
    ],
    [
      "3776697713835641",
      "158802844398077608725829"
    ],
    [
      "4074831617548741",
      "168575528808112484387629"
    ],
    [
      "8859168458759941",
      "697209736364853202214829"
    ],
    [
      "26766265120243039/4",
      "3273410326945663582112807/8"
    ],
    [
      "332381580514018741/81",
      "123794864495365886795230181/729"
    ],
    [
      "8372672447080341",
      "628499827445905949988429"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber 089 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -68/89. A(u) = (-22234134433116285910092921277275)+(110922098753374475218610657758560)*u^1+(-2238835299613016412598155663300)*u^2+(-668291528103291204503696355619440)*u^3+(-1228382975540396813143327213163490)*u^4+(-1305046090974099330756097785833760)*u^5+(-730165036282222351438425309089220)*u^6+(-115184111329442799000859184518320)*u^7+(-6764206525058961744342207836427)*u^8 ; B(u) = (43437345573952011026007762008660031135091893750)+(-272653378639351264146660169730601346460528493600)*u^1+(274681422677639502079180012059258901947525880524)*u^2+(834095500002542547490222678572818893299399330960)*u^3+(-1446190337475684620303144196583106816058012416310)*u^4+(4817627385439859969516421772920670917055554020640)*u^5+(25741889274749849075903353261318279749128611700520)*u^6+(35618011375130752993917833852707386766033265770656)*u^7+(22305750380707309697813397419957100040991477108250)*u^8+(7478862206043393848507380663800830742000201885120)*u^9+(1781657943811020988023763350544021409036455304940)*u^10+(219381245128735196370337431665053397514385169040)*u^11+(9838320680474420154415220509114637635180818246)*u^12. Rank exactly 23: 23 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 6 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e8 on the p1 cluster, exact halving), canonical height Gram det != 0; Bober analytic rank bound 23 (kappa=2.8, GRH), rank is exactly 23.",
  "created_at": "2026-10-21 13:23:52",
  "updated_at": "2026-10-21 13:23:52"
}