{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "5WGZPE6249"
    },
    {
      "field": "uses",
      "target": "M5196PBDCA"
    }
  ],
  "body": "\n## Summary\n\nA Weierstrass equation is a cubic equation $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ with coefficients in a field.\nWhen its discriminant is nonzero, it defines an [[elliptic curve|427N6S6QQW]], with the point at infinity as the origin.\nEvery elliptic curve can be written in this form, so Weierstrass equations are the standard way to compute with elliptic curves.\nThe name comes from the differential equation of the [[Weierstrass elliptic function|361NAK68QK]], which yields such an equation over $\\C$.\n\n## Explanations\n\n### L1 human\n\nA Weierstrass equation says that $y^2$, up to some correction terms, equals a cubic polynomial in $x$.\nThe simplest case is $y^2 = x^3 + ax + b$.\nOver the real numbers its graph is symmetric about the $x$-axis, and when it is smooth it has one or two pieces.\nFor example, $y^2 = x^3 - x$ is a smooth curve, while $y^2 = x^3$ has a sharp cusp at the origin and does not count as an elliptic curve.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14H52",
      "11G05"
    ],
    "name": "Weierstrass equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Elliptic_curve"
      }
    ],
    "tag": "Z3C8B6QTS6"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Z3C8B6QTS6/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/Z3C8B6QTS6.md",
  "msc": [
    "14H52",
    "11G05"
  ],
  "name": "Weierstrass equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z3C/Z3C8B6QTS6.md",
  "status": null,
  "tag": "Z3C8B6QTS6",
  "uri": "m2://Z3C8B6QTS6",
  "url": "https://tamnd.github.io/math2/wiki/weierstrass-equation/",
  "writing": {
    "tag": "Z3C8B6QTS6",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}