{
  "id": 4204,
  "curve_key": "77819427678490434706729024176945574076078241:-688424833110372134023666925453715122757437877307559065368344306289",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-1621238076635217389723521337019699459918297",
    "796788001285152932898167954720217974205576571662448287559822521"
  ],
  "rank_lower_bound": 26,
  "torsion": [],
  "naive_height": 303.19453399195146,
  "faltings_height": 23.07886663679182,
  "conductor": "2967355291463937048383780024918256242192132464440126497308674951954423661079347962552287794552076810",
  "bad_primes": [
    "2",
    "5",
    "7",
    "11",
    "17",
    "19",
    "23",
    "37",
    "89",
    "21481",
    "5182753318937",
    "1414951979967742058549850114727406900636749336334406032440944959867012917"
  ],
  "discriminant": "-1542206512749950257160613497475126110089398017151089574666554577062896021963744582072395315009675056967342713184844513280000000",
  "regulator": "11523870156431655305799520657.5457908049794392665106363984524610965928710173094582609543325",
  "points": [
    [
      "-1440789547016287468889",
      "-11905988733267710368254710441556"
    ],
    [
      "765760199599265544261",
      "2083632232564070894563948124394"
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    [
      "975933547599719656871",
      "12003818545931977567020299481004"
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    [
      "4572794775960877378311",
      "-298332708485244129025386323707956"
    ],
    [
      "473399742844765051111",
      "11635577283624067175168138830444"
    ],
    [
      "729531088583804123911",
      "1520610118111001832546071504844"
    ],
    [
      "-1546225711825448788001/9",
      "-883296391283205333372019555042012/27"
    ],
    [
      "-86558494715533493849",
      "30601820211345001685061324334764"
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      "770920963752770046111",
      "2261561200093985862977747850444"
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    [
      "-901540235052476125609",
      "-39059572166491764933461061876676"
    ],
    [
      "806781503045277844231",
      "3732864827539558589899132911564"
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    [
      "-1414552313080694895289",
      "-16113781577856248027144028479956"
    ],
    [
      "46748149417492366326679/169",
      "-42231169188762851896564375461910652/2197"
    ],
    [
      "866652304409985184679",
      "-6532142583045928164622829238356"
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    [
      "-164476027387783990003601/2809",
      "-4445208899243012403994559063645023812/148877"
    ],
    [
      "772775784140916732071",
      "-2328643912905723847198220168276"
    ],
    [
      "1276412917294981283111",
      "-28407550283856303098297314345556"
    ],
    [
      "1118275965621592787751",
      "19551125830090584861924565712684"
    ],
    [
      "-1201764831556674119609",
      "31772623670590533956019947809324"
    ],
    [
      "894368653327463393611",
      "-7886999274077429897151442541556"
    ],
    [
      "1355020626118526286985471/2025",
      "-309790344293392174631976992703318116/91125"
    ],
    [
      "783352704491052676711",
      "-2735775269082494490428214588756"
    ],
    [
      "205568702699846001975529/289",
      "9166235107510577102146735556686022/4913"
    ],
    [
      "494124134865546019411",
      "-10786094113668252616544478291156"
    ],
    [
      "7641809246567536965617791/11881",
      "5806746306245217647173319896360734876/1295029"
    ],
    [
      "447902079597334380711",
      "-12668407203745807444570670236756"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -318/133 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = 1840758517133685517328T^8 - 28551969868039231295840T^7 - 1772998693252999313402080T^6 - 759387847760867496814400T^5 - 598922555435041558626640T^4 - 544800986643777256848080T^3 + 926816232306216461908040T^2 + 288493294443742559437240T - 230970134041531372322875,\nA6 = 325113825546664597334681997252736T^12 + 13439521326531181915894800771978880T^11 + 138327069120450496577609764257248960T^10 + 3518007078129231511599849398766400T^9 + 1769321142185421847687389290390981520T^8 + 1594947821477300217886357939071742368T^7 - 1479100347162091017597427825166729760T^6 - 1235366662137150944785202891205624960T^5 + 731032719531350653488345005675279040T^4 + 467848968404338281084037609996168720T^3 - 249593152301734103394166433543833544T^2 - 84246673758026015337923892531146920T + 42769382258496484819391456728787350.\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": "2026-10-13 03:41:19",
  "updated_at": "2026-10-13 03:41:19"
}