{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], let $E_4$ and $E_6$ be the normalized [[Eisenstein series|EYREVY31BN]] of weights $4$ and $6$, and let $\\Delta = (E_4^3 - E_6^2)/1728$.\nThe *$j$-invariant* is the function $j \\colon \\mathbb{H} \\to \\C$ given by\n$$j(\\tau) = \\frac{E_4(\\tau)^3}{\\Delta(\\tau)} = \\frac{1728\\, E_4(\\tau)^3}{E_4(\\tau)^3 - E_6(\\tau)^2}.$$\n\n## Remarks\n\nThe [[cusp form|A50Y012RBP]] $\\Delta$ has no zeros on $\\mathbb{H}$, so $j$ is holomorphic on $\\mathbb{H}$.\nBoth $E_4^3$ and $\\Delta$ have weight $12$, so $j(\\gamma \\tau) = j(\\tau)$ for every $\\gamma$ in the [[modular group|VEKV1EM77B]].\nWith $q = e^{2 \\pi i \\tau}$, $j(\\tau) = q^{-1} + 744 + 196884 q + \\cdots$, so $j$ is a [[meromorphic|0ZZ7ZYDR8W]] modular function with a simple pole at infinity.\nExample: $E_6(i) = 0$, so $j(i) = 1728$; and $E_4(e^{2 \\pi i/3}) = 0$, so $j(e^{2 \\pi i /3}) = 0$.\nNon-example: $j$ is not a [[modular form|GWKN5VG78A]] of weight $0$, because it is unbounded as $\\operatorname{Im} \\tau \\to \\infty$.\nFor the elliptic curve $y^2 = x^3 + ax + b$, the $j$-invariant is $1728 \\cdot 4a^3 / (4a^3 + 27b^2)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FEC2XH822R",
    "kind": "definition",
    "msc": [
      "11F03",
      "14H52"
    ],
    "name": "j-invariant",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/J-invariant"
      }
    ],
    "tag": "K8BK5SKY1Y",
    "uses": [
      "EYREVY31BN",
      "A50Y012RBP",
      "X6XGGH5FPW",
      "VEKV1EM77B",
      "0ZZ7ZYDR8W"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/K8BK5SKY1Y/",
  "links": [
    {
      "field": "defines",
      "target": "FEC2XH822R"
    },
    {
      "field": "uses",
      "target": "EYREVY31BN"
    },
    {
      "field": "uses",
      "target": "A50Y012RBP"
    },
    {
      "field": "uses",
      "target": "X6XGGH5FPW"
    },
    {
      "field": "uses",
      "target": "VEKV1EM77B"
    },
    {
      "field": "uses",
      "target": "0ZZ7ZYDR8W"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/K8BK5SKY1Y.md",
  "msc": [
    "11F03",
    "14H52"
  ],
  "name": "j-invariant",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K8B/K8BK5SKY1Y.md",
  "status": null,
  "tag": "K8BK5SKY1Y",
  "uri": "m2://K8BK5SKY1Y",
  "url": "https://math2.tamnd.com/wiki/j-invariant-definition/",
  "writing": {
    "tag": "K8BK5SKY1Y",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}