Crystallography & Symmetry
patterns derived from their groups
Concept

Class number — where it appears

How many inequivalent lattices share a given metric determinant, counted up to a change of basis by an integer matrix of determinant one. It is finite, it is computed by enumerating reduced forms, and it is finer than anything a congruence can measure — which is what the genus counts instead.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

How many different lattices share a determinant. One bar per determinant: the number of inequivalent integral lattices whose metric has that determinant, which is the class number of the corresponding discriminant. Area does not decide shape — at determinant 1 and 2 there is one lattice each, and by 11 there are four — and the count does not grow steadily either. Each bar is computed twice: once by enumerating the reduced forms directly, and once by reducing every form in a box and collecting the distinct results, which is a search followed by an algorithm rather than a search over answers. The two agree at every bar.

How many lattices share a determinant

Area does not decide shape. The number of inequivalent lattices whose metric has a given determinant is a class number, computed by enumerating reduced forms — and checked by reducing every form in a box and counting what comes back distinct.

lattices · Moduli Three lattices no congruence can separate. The three reduced forms of discriminant minus twenty-three, with the integers each represents. The principal form represents one and the others do not; the others represent two and it does not. So they are genuinely different lattices — and they represent exactly the same residues modulo twenty-three, so they are in one genus and no congruence condition of any kind distinguishes them.

Lattices that agree at every prime

Counting the plane lattices with a given metric determinant is a class number. Above it sits a coarser count — the genus, which is what congruences can see — and for most small determinants the two agree. At discriminant minus twenty-three they part: three lattices representing exactly the same residues modulo everything, and different integers. No argument modulo any number can tell them apart, and they are not the same lattice.

lattices · Moduli

Named alongside it

The objects these essays reach for when they reach for this one.

Theta seriesBinary quadratic formDiscriminantEquivalence classGram matrixHexagonal latticeLatticeLattice reductionNormal formQuadratic form

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