Proving that this expression can never be a perfect cube

Question

I want to prove that $\dfrac{3m^2+1}{4}$ can never be a perfect cube. Here $m$ is an odd number greater than $1$. Is there a simple way to do that? I saw other answers proving some expression cannot be perfect squares using modulo operators. Could that be used here in this case?

Answer 1

from $3m^2 + 1 = 4 w^3$ we get the Mordell curve $$ (36m)^2 + 432 = (12w)^3 $$

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However:

jagy@phobeusjunior:~$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath Version 6.9, Release Date: 2015-10-10                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage: E = EllipticCurve([0,0,0,0,-432])
sage:  E.integral_points()
[(12 : 36 : 1)]
sage: quit
Exiting Sage (CPU time 0m0.50s, Wall time 1m3.94s).
jagy@phobeusjunior:~$