{
  "id": 5181,
  "curve_key": "708683947367811731173349371425:-591733034740327069740130988315053191109160625",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-14764248903496077732778111905",
    "684876197616122984742933036421115128682097"
  ],
  "rank_lower_bound": 22,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 206.19962149675342,
  "faltings_height": 15.039648156008814,
  "conductor": "854719095669670901420795176752574947323841009617302550504741228442740450",
  "bad_primes": [
    "2",
    "5",
    "19",
    "23",
    "1747",
    "995387",
    "299079203",
    "3958651592872630315675970842186160379851222074109"
  ],
  "discriminant": "3342844948089679511007802042557695695165165440319624207715121718363785236399104000000",
  "regulator": "65397197186819225806614.18614656891493542329311510114723075794910681594182845426347",
  "points": [
    [
      "-11336790594577326729/85849",
      "-14487438596926275979795431180/25153757"
    ],
    [
      "-78766284252841",
      "-1165815190645163018780"
    ],
    [
      "-18782945860664729/225",
      "3900473103366240580344236/3375"
    ],
    [
      "-75146366115081",
      "1170472634271157603940"
    ],
    [
      "16990407331015",
      "662518129111099411348"
    ],
    [
      "-38374306093956009/529",
      "-14262735619434171860919380/12167"
    ],
    [
      "-132365451636621",
      "565710988214161036260"
    ],
    [
      "-407492410499762945/3481",
      "184732835733824743466498124/205379"
    ],
    [
      "-127858967554121",
      "694543230979797436260"
    ],
    [
      "-6121556431955801/81",
      "852971195162324598713540/729"
    ],
    [
      "-94077767242841",
      "-1114098965377526578380"
    ],
    [
      "-112081901131881",
      "-965229929890208767260"
    ],
    [
      "171987950472892531/2809",
      "15203198531683742099261940/148877"
    ],
    [
      "-627997690565291177/6561",
      "-587268526220181428563045340/531441"
    ],
    [
      "207981690583344871/3249",
      "-8432862992676483445821020/185193"
    ],
    [
      "-1009403110104689/9",
      "26037449749020106135820/27"
    ],
    [
      "-87956262563337",
      "1141502542403944163940"
    ],
    [
      "64307310558455",
      "36951988345069243556"
    ],
    [
      "-66151881393211/16",
      "55272002336978391748415/64"
    ],
    [
      "83369798777119",
      "182886454170385466140"
    ],
    [
      "57745876752256639/441",
      "9245984600015853065877460/9261"
    ],
    [
      "134362303648119",
      "1061504097616933512740"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber 133 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -27/98. A(u) = (-98446928369392297987858107)+(753656616847155064495894056)*u^1+(-1304662621155467165949437316)*u^2+(-4515151930070151665108601288)*u^3+(-4977701063274588505383095730)*u^4+(-3216158287288357141650052008)*u^5+(-379643944497665001753404196)*u^6+(387597384378623366437498056)*u^7+(-52947472367727871375145067)*u^8 ; B(u) = (455529334666944868668501131133770300694)+(-4406091141778868895239142866363000327928)*u^1+(8711059880694673976260419900526416998364)*u^2+(-3793179221642104157592128359195176701880)*u^3+(24216919834250864412914656434287671696410)*u^4+(200113292396607048670446102942302942345616)*u^5+(266455804027297683905238442306815065604168)*u^6+(107441306099609449962047233442503188641616)*u^7+(-2582417909382033421927144160479025688150)*u^8+(428474205328592695582331592965376042600)*u^9+(2199423551161536880161124686788281724316)*u^10+(-1449946611991997592459259221798535014552)*u^11+(194470684143269906473287684559611258726)*u^12. Rank 22 (exactly): 22 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 5 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e6, exact halving), 22x22 canonical height det = 6.54e22 != 0; Bober analytic rank bound 22 (kappa=2.8, GRH), root number +1.",
  "created_at": "2026-10-21 08:08:27",
  "updated_at": "2026-10-21 08:08:27"
}