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  "body": "\n## Summary\n\nAn Eichler order is an [[order|1ZBX4TKVZ9]] in a [[quaternion algebra|ZWXW0T3EAC]] that is the intersection of two [[maximal orders|E5DH1CSW1N]].\nEvery maximal order is an Eichler order, as the intersection of itself with itself.\nIn $M_2(\\Q)$ the standard Eichler order of level $N$ is the ring of integer matrices whose lower left entry is divisible by $N$.\nIts elements of determinant $1$ form the congruence subgroup $\\Gamma_0(N)$.\nEichler orders in indefinite and definite quaternion algebras give Shimura curves and the Brandt matrices used to study modular forms.\n\n## Explanations\n\n### L1 human\n\nIn the rational $2 \\times 2$ matrices, the integer matrices form a maximal order, and so does each of its conjugates.\nIntersecting two of these gives a slightly smaller ring with a congruence condition.\nFor example, the integer matrices $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ with $c$ divisible by $N$ form such an intersection.\nThis ring is behind the group $\\Gamma_0(N)$, which is basic in the theory of modular forms.\nEichler orders carry the same idea to other quaternion algebras.\n",
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