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The Igusa-Clebsch invariants of a genus 2 curve are weighted projective invariants associated to the geometric isomorphism class of the curve. They are denoted $I_2, I_4, I_6, I_{10}$, where the subscript indicates the weight.

For genus 2 curves over $\Q$ we may assume that the Igusa-Clebsch invariants are integral. We can then normalize them by removing a factor of $d^k$ from $I_{2k}$, where $d$ is the largest positive integer for which the invariants remain integral after this normalization. We can also make the first nonzero $I_{2k}$ with $k$ odd positive by multiplying each $I_{2k}$ by $(-i)^k$.

In the LMFDB, Igusa-Clebsch invariants for genus 2 curves over $\Q$ are always normalized in this way, which means that their values may differ from those returned by the corresponding functions in Sage or Magma; however, they are equivalent as $\overline\mathbb{Q}$-points in weighted projective space and identify the same geometric isomorphism class.

The exact definition of the Igusa-Clebsch invariants is as follows. Let $X$ be a curve of genus $2$, which we may assume to be defined by an equation $y^2 = f(x)$ with $f$ a sextic in $x$. Let $c$ be the leading coefficient of $f$, fix an ordering of the roots $x_1,\ldots,x_6$ of $f$ in its splitting field, and let $(i j)$ denote the difference $x_i-x_j$. Then \[ I_2 = c^2 \sum (1 2)^2 (3 4)^2 (5 6)^2,\qquad\qquad\qquad\qquad\qquad\quad \\ I_4 = c^4 \sum (1 2)^2 (2 3)^2 (3 1)^2 (4 5)^2 (5 6)^2 (6 4)^2,\qquad\qquad\quad\ \ \\ I_6 = c^6 \sum (1 2)^2 (2 3)^2 (3 1)^2 (4 5)^2 (5 6)^2 (6 4)^2 (1 4)^2 (2 5)^2 (3 6)^2, \\ I_{10} = c^{10} \prod (1 2)^2.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \\ \] Here each sum and product runs over the distinct expressions obtained by applying a permutation to the index set $\{1, \ldots, 6\}$. More precisely, it is obtained by averaging over the $S_6$-Galois orbit of the given expression. Note that $I_{10}$ is simply the discriminant of the sextic polynomial $f$.

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  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2022-01-25 20:47:39
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