{
  "id": 6078,
  "curve_key": "2070167314514344720145721909517455306832677603841:-2978572468846222660186233064736023175368247524594001451986230778567151361",
  "ainvs": [
    "1",
    "0",
    "0",
    "-43128485719048848336369206448280318892347450080",
    "3447421838942387338178506897552142828531443641216593953476007557350400"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    7
  ],
  "family": null,
  "families": [],
  "naive_height": 333.7551416880273,
  "faltings_height": 25.17380728135887,
  "conductor": "345217887147913916505132892259670",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "29",
    "37",
    "47",
    "59",
    "2161",
    "1367187347193435109"
  ],
  "discriminant": "-250463013200817510497607244191900606194505183108619801456921428150838821028931279486212122242380550623521717585203980800000000000000",
  "regulator": "38132769.871662020762602294298350393336326303001611",
  "points": [
    [
      "3942048502533368651765652960/6241",
      "235529992257727720768173202113609728398080/493039"
    ],
    [
      "952738760937391300234253760/7921",
      "160530299309499352776516115555478107680/704969"
    ],
    [
      "1052651291828949943810062229440/8779369",
      "7621503385292343232534519185120762937440/26013270347"
    ],
    [
      "4374060507384056055364554240/36481",
      "4438845089561306351797362797239261920/6967871"
    ],
    [
      "74892898845579584388306063470635860/624556703521",
      "3867744152732830052078520020365828824021883869180/493580292668907569"
    ],
    [
      "32529425855908735368240449330161050/22084040449",
      "5811571220685218557713606468665058529890464193853110/3281842999004543"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "From znan2's published specialization at t=12925/1089144 of Kubert's Z/7Z Tate family y^2+(1-c)*x*y-b*y=x^3-b*x^2, with b=t^3-t^2 and c=t^2-t. The source records a height-ordered search, PARI/GP rank bounds [6,6], and saturation to 100. Its six points are reproduced here on the same global minimal model. The specialization-to-model identity and all six point coordinates were checked exactly. Their independence was independently certified by reduction homomorphisms to F_5 at seven good primes, giving a matrix of rank 6; the rational torsion subgroup is Z/7Z. The complete bad-prime list was verified from the minimal discriminant. Source: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/6d9049fb3a1357d0199aada9d2c95c4de8fe2369/submission/board/7-6.ec",
  "created_at": "2026-10-29 00:36:19",
  "updated_at": "2026-10-29 00:36:19"
}