{
  "id": 4227,
  "curve_key": "3270101225933209711747078530732422351849296:-5644346928388036224016901558883866606153440053552494868030161344",
  "ainvs": [
    "0",
    "1",
    "0",
    "-68127108873608535661397469390258798996860",
    "6532808944893560444441297397602161282271941646259375386480400"
  ],
  "rank_lower_bound": 27,
  "torsion": [],
  "naive_height": 293.6801845383685,
  "faltings_height": 22.408251231420877,
  "conductor": "207479491880304957471326786939605942587517259346001379219559248730591334381853263329044682907789019286360",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "19",
    "31",
    "43",
    "103",
    "20231",
    "23899",
    "3569885381",
    "9126168707",
    "1544303695164929",
    "3892264626982588260567838467320645992725385851005177"
  ],
  "discriminant": "1799987296794708805147241090174322551863878104622532848801918616074317870058655341928888242110152543001229725760064116000000",
  "regulator": "141743139184419384275617828872.80119308433360541022928556721936010607230067793483675952805838",
  "points": [
    [
      "242387149450676369440",
      "2064046792180316916479018260140"
    ],
    [
      "214981763702437237975",
      "-1350025636508247323583061245870"
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    [
      "203350902983094556480",
      "1043064212814658617469432011900"
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    [
      "5104108126103493955",
      "2487009004145180083387254870000"
    ],
    [
      "280021453218802005760",
      "-3068029187961607987849487581980"
    ],
    [
      "-232166661576233530370",
      "-3136168659383722097604615815550"
    ],
    [
      "121159210339899257890",
      "239051107760016009690251766810"
    ],
    [
      "798201115786005316330",
      "-21464099736007339740188135274750"
    ],
    [
      "12535116125798032387180/169",
      "-3018558484826354943547594398476220/2197"
    ],
    [
      "180338519522469952210",
      "334414427335281314129933093430"
    ],
    [
      "120020344916946664288",
      "291631426836744781701921888756"
    ],
    [
      "122137602093176362930",
      "-184191125647151756975180212650"
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    [
      "318034406781641660320",
      "-4127216431458214423535506703340"
    ],
    [
      "9985924876717556898417580",
      "-31556035917386929429800826970014618500"
    ],
    [
      "122832719276167294380",
      "-133625807010851586517607864700"
    ],
    [
      "246310779922287622675",
      "-2166978635529774426600988889820"
    ],
    [
      "-275229689796252339890",
      "-2105797373298935596001947138830"
    ],
    [
      "7463723375425311616378330/66049",
      "8938029918597747932555077442761065330/16974593"
    ],
    [
      "-298190149193808271670",
      "-577372796430234948807228798750"
    ],
    [
      "-83053025459624470550",
      "-3408531630369254037192501910530"
    ],
    [
      "-97611108208497964052",
      "-3500391390483684957435386506596"
    ],
    [
      "-281563015054303308032",
      "1842065697402283527899876994276"
    ],
    [
      "-108172787006488986828788/529",
      "-41995361347862145501558120365597772/12167"
    ],
    [
      "371912723358334383820",
      "-5712975912515585453996385224340"
    ],
    [
      "455389096735081309915870/2209",
      "-116051278744402913250171342006106050/103823"
    ],
    [
      "32841105338499687091345/64",
      "5288540213301408773229821751275175/512"
    ],
    [
      "187367314434910605310",
      "588071517821208175332157606530"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = 433/670 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -5359397556934326557952T^8 - 135465106160634648571008T^7 - 1009560857120332086707244T^6 + 302645697651446768292456T^5 + 267102270709279813507320T^4 - 18710244598364193145536T^3 + 1090445903163236801932416T^2 - 91760564814653904426432T - 567824854400092808778672,\nA6 = 156065240689863717949223596360704T^12 + 4827291258544591401284174444461824T^11 + 48077643204571887106158625640791056T^10 + 215456560411136671924188808659259680T^9 + 998209464630408701484689228105999700T^8 - 909644941490611250119006829109042720T^7 - 2569542697812503924009268807282804720T^6 + 1241192156761670225595583604913427680T^5 + 2544459298635871164419215332497359200T^4 - 725030770745235523201629773993683200T^3 - 1216067410812944483775200857080372480T^2 + 159116305290413916148713260094977280T + 238920668104630371120386669007676800.\nIt is a 2-neighbour of the fibration given in curve#4205 (itself a 2-neighbour of the fibration 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": "2026-10-14 17:52:07",
  "updated_at": "2026-10-14 17:52:07"
}