{
  "id": 5292,
  "curve_key": "2500698221198615565708594115921:-3959820227421916460207817829337875974494408281",
  "ainvs": [
    "1",
    "0",
    "0",
    "-52097879608304490952262377415",
    "4583125263219806376343155128951999318985225"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 209.98505562370457,
  "faltings_height": 15.283038663352981,
  "conductor": "9129236490831315872895122536046465273520242669492015983172432970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "59",
    "79",
    "157433",
    "14705177776139660808281",
    "2563748174699285461305899480653"
  ],
  "discriminant": "-24352389663777816211849952728040007004618067230556682634967274681504956446801920000000",
  "regulator": "1669595456800475113.56295257465749618801874580558269183766113359589542425312660",
  "points": [
    [
      "293778295466230",
      "3825262795341762464485"
    ],
    [
      "53710572837430",
      "1392789958465093117285"
    ],
    [
      "-662250877166354714/15625",
      "5061234590767314972052570841/1953125"
    ],
    [
      "-102363037136954",
      "2973792739953078463477"
    ],
    [
      "24968680116030",
      "1816005582582365057685"
    ],
    [
      "719332773422042710/5041",
      "-83606805280035859567057645/357911"
    ],
    [
      "303786043704674800/1681",
      "71297229468781097989409435/68921"
    ],
    [
      "646537487708659/4",
      "-4964666233211835808369/8"
    ],
    [
      "1962400895079562",
      "86368893553857612788941"
    ],
    [
      "158463703288630",
      "-553754549400075874715"
    ],
    [
      "64385558925952990/841",
      "24911669792593258923009425/24389"
    ],
    [
      "-4229105840309930/49",
      "-996276715996452556611245/343"
    ],
    [
      "359067254299210",
      "5671928059891779799885"
    ],
    [
      "45308254339642870/289",
      "-2547095283064315573037995/4913"
    ],
    [
      "179572529942470/9",
      "50883255067500975213895/27"
    ],
    [
      "1399812329497908205/13456",
      "-839470446113015147556640765/1560896"
    ],
    [
      "28936951979446",
      "1760625356865944352997"
    ],
    [
      "88972480901230",
      "807566008917292754485"
    ],
    [
      "88635867717430",
      "813471514088942365285"
    ],
    [
      "133939113971446",
      "89482390183738421797"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber of the x1092 K3 fibration family (file 157_x1092_t0_0071.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -1/51. A(u) = (-133266803352683433867)+(-34429351041428677740648)*u^1+(-5159465855422129650830724)*u^2+(-199867014403654025300872824)*u^3+(-3674407941552005191068242130)*u^4+(-39203515617102192576477557016)*u^5+(-242832298117300513973865560484)*u^6+(-697681340718393915812184817032)*u^7+(-718622366318760700413751004667)*u^8 ; B(u) = (539316574939092550969221207750)+(261194186737739667071325677113080)*u^1+(52894082157207048461637283455819900)*u^2+(4127823374881053534231122609192191800)*u^3+(235551250432437047361240270551409681450)*u^4+(8604812583907478066799326365826483591280)*u^5+(187613055547997482853550846684987195861000)*u^6+(2452920875396144599562109396854511343838640)*u^7+(19341237044985803273673258128017269067453050)*u^8+(91311139145529018029385201123473804359568280)*u^9+(253761827381667465095652734751449752291673916)*u^10+(383945461130653860983197400998299471308869656)*u^11+(243804338752456642528814405356804610148815574)*u^12. Rank exactly 20: 20 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 3 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e6, exact halving), canonical height Gram det != 0; Bober analytic rank bound 20 (kappa=2.8, GRH), rank is exactly 20.",
  "created_at": "2026-10-21 12:16:46",
  "updated_at": "2026-10-21 12:16:46"
}