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# A Genus 43 Curve - Differentials

Mar 3, 2016

Consider the genus $43$ Riemann surface $X$ obtained from the desingularization and compactification of the plane algebraic curve

$C: f(x,y) = y^3 \big((y^4 - 16)^2 - 2x\big) - x^{12}.$

The space of holomorphic differentials on $X$ is spanned by the following $43$ differentials:

\begin{aligned} y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^5/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^6/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^7/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^8/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ y^9/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^5/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^6/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^7/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ xy^8/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ x^2y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2) \\ x^2y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^2y^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^2y^5/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^2y^6/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^2y^7/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^3y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^3y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^3y^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^3y^5/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^3y^6/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^4y/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^4y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^4y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^4y^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^4y^5/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^5y/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^5y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^5y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^5y^4/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^6y/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^6y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^6y^3/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^7y/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^7y^2/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^8/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^8y/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ x^9/(11y^{10} - 224y^6 - 6xy^2 + 768y^2)\\ \end{aligned}