{
  "id": 236,
  "curve_key": "1928124692225363784115008:-1781647907844700462982130903918099456",
  "ainvs": [
    "0",
    "0",
    "0",
    "-40169264421361745502396",
    "2062092485931366276599688546201504"
  ],
  "rank_lower_bound": 16,
  "naive_height": 167.75577030103472,
  "faltings_height": 12.01316038217705,
  "conductor": "2329523824211488655524107381802033064249929281525454944",
  "bad_primes": [
    "2",
    "3",
    "11",
    "19",
    "31",
    "37",
    "79",
    "163",
    "1187",
    "2207494092079386051158169308866468641719"
  ],
  "discriminant": "2311256994351959597357327359870479823158106106819333943959339451297792",
  "regulator": "1575306794443644.152922817277863822046611524738222651298257620999230147",
  "points": [
    [
      "263473897774",
      "98835728473299268"
    ],
    [
      "1694406292782",
      "2190584624758308540"
    ],
    [
      "107788063883300184/16129",
      "35372203232200059004682544/2048383"
    ],
    [
      "177077545644550110/1026169",
      "17045499838668802241704524/1039509197"
    ],
    [
      "698820360270",
      "-561480046717032828"
    ],
    [
      "6318338595346081/8100",
      "-486498513485524345029521/729000"
    ],
    [
      "5853814809142294/34969",
      "35076549330988415699084/6539203"
    ],
    [
      "316593635550",
      "-145180658240849052"
    ],
    [
      "12138887762392620/63001",
      "607416600498386153991228/15813251"
    ],
    [
      "182224971549",
      "-28163823213797307"
    ],
    [
      "25789670918080633836/5058001",
      "-130868856464406362882549387748/11375444249"
    ],
    [
      "63166954843761178486/375623161",
      "-57635319338620210678550548852/7279952483341"
    ],
    [
      "431450722493976/961",
      "-8132585828692923037728/29791"
    ],
    [
      "667137117769/4",
      "-11108990118817691/8"
    ],
    [
      "260509642146528246/1428025",
      "-48423597185946377910988044/1706489875"
    ],
    [
      "1244240230476587934/3545689",
      "-1178907764805735000176562444/6676532387"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥16. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-7,-4), cleared tuple [-210, -120, 330, 875, -864, -11], shift t=606/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 16 via 16 independent points from a quartic point search + Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-26 06:57:26",
  "updated_at": "2026-07-01 22:38:40"
}