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edited Apr 19 at 1:04

Here are two other approaches to the formula, one of which may satisfy you, depending on your tastes:

Roquette's Analytic Theory of Elliptic Functions over Local FieldsAnalytic Theory of Elliptic Functions over Local Fields includes a self-contained discussion of this formula in Section 3 on pages 23 to 29. Roquette's approach, though, is (non-Archimedean) analytic and also uses a bit of the theory of genus-1 function fields--not really low-level.
In the paper where he introduces the tau function, "On certain arithmetical functions"On certain arithmetical functions," Ramanujan sketches a direct proof of something related to $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$, namely $\wp'' = 6\wp^2 + \frac{a_4}{2}$ (Equation (21) in that paper). One can find some further details of the computations in Chapter 4 ("Eisenstein Series") of Berndt's Number Theory in the Spirit of RamanujanTheory in the Spirit of Ramanujan, but Berndt leaves part of the work as an exercise for his readers. A more comprehensive treatment, generalizing Ramanujan's approach, is in the first chapter of Developmentfirst chapter of Elliptic Functions According to RamanujanDevelopment of Elliptic Functions According to Ramanujan by Venkatachaliengar and Cooper. The authors reach $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$ at the bottom of page 11. Perhaps by reducing the generality of Venkatachaliengar and Cooper's intermediate formulas, one could produce a minimal low-level proof that fills in the details of Ramanujan's sketch.

Here are two other approaches to the formula, one of which may satisfy you, depending on your tastes:

Roquette's Analytic Theory of Elliptic Functions over Local Fields includes a self-contained discussion of this formula in Section 3 on pages 23 to 29. Roquette's approach, though, is (non-Archimedean) analytic and also uses a bit of the theory of genus-1 function fields--not really low-level.
In the paper where he introduces the tau function, "On certain arithmetical functions," Ramanujan sketches a direct proof of something related to $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$, namely $\wp'' = 6\wp^2 + \frac{a_4}{2}$ (Equation (21) in that paper). One can find some further details of the computations in Chapter 4 ("Eisenstein Series") of Berndt's Number Theory in the Spirit of Ramanujan, but Berndt leaves part of the work as an exercise for his readers. A more comprehensive treatment, generalizing Ramanujan's approach, is in the first chapter of Development of Elliptic Functions According to Ramanujan by Venkatachaliengar and Cooper. The authors reach $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$ at the bottom of page 11. Perhaps by reducing the generality of Venkatachaliengar and Cooper's intermediate formulas, one could produce a minimal low-level proof that fills in the details of Ramanujan's sketch.

Here are two other approaches to the formula, one of which may satisfy you, depending on your tastes:

Roquette's Analytic Theory of Elliptic Functions over Local Fields includes a self-contained discussion of this formula in Section 3 on pages 23 to 29. Roquette's approach, though, is (non-Archimedean) analytic and also uses a bit of the theory of genus-1 function fields--not really low-level.
In the paper where he introduces the tau function, "On certain arithmetical functions," Ramanujan sketches a direct proof of something related to $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$, namely $\wp'' = 6\wp^2 + \frac{a_4}{2}$ (Equation (21) in that paper). One can find some further details of the computations in Chapter 4 ("Eisenstein Series") of Berndt's Number Theory in the Spirit of Ramanujan, but Berndt leaves part of the work as an exercise for his readers. A more comprehensive treatment, generalizing Ramanujan's approach, is in the first chapter of Development of Elliptic Functions According to Ramanujan by Venkatachaliengar and Cooper. The authors reach $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$ at the bottom of page 11. Perhaps by reducing the generality of Venkatachaliengar and Cooper's intermediate formulas, one could produce a minimal low-level proof that fills in the details of Ramanujan's sketch.

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created Apr 18 at 19:57

Here are two other approaches to the formula, one of which may satisfy you, depending on your tastes:

Roquette's Analytic Theory of Elliptic Functions over Local Fields includes a self-contained discussion of this formula in Section 3 on pages 23 to 29. Roquette's approach, though, is (non-Archimedean) analytic and also uses a bit of the theory of genus-1 function fields--not really low-level.
In the paper where he introduces the tau function, "On certain arithmetical functions," Ramanujan sketches a direct proof of something related to $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$, namely $\wp'' = 6\wp^2 + \frac{a_4}{2}$ (Equation (21) in that paper). One can find some further details of the computations in Chapter 4 ("Eisenstein Series") of Berndt's Number Theory in the Spirit of Ramanujan, but Berndt leaves part of the work as an exercise for his readers. A more comprehensive treatment, generalizing Ramanujan's approach, is in the first chapter of Development of Elliptic Functions According to Ramanujan by Venkatachaliengar and Cooper. The authors reach $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$ at the bottom of page 11. Perhaps by reducing the generality of Venkatachaliengar and Cooper's intermediate formulas, one could produce a minimal low-level proof that fills in the details of Ramanujan's sketch.