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I've heard that integral solutions to equations of the form $y^2 = x^3 + a$ for $a\in\mathbb{Z}$ yield binary cubic forms of discriminant related to $a$.

Where can I find a reference for this?

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This connection is outlined on page 246 of Mordell's book "Diophantine Equations". More explicit details can be found at https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB62147F66768E3777082ABC1DBFCE27/S1461157015000182a.pdf/mordells_equation_a_classical_approach.pdf

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