Uniform bound forcing total balance

Determine a computable uniform bound on the prime ell such that every heavenly elliptic curve defined over a cubic number field is totally balanced at ell.

Background

The paper proves that every heavenly elliptic curve over a cubic number field is balanced once ell exceeds the explicit bound 73. A totally balanced curve is a more restrictive type of balanced curve for which the two exponents in the mod-ell Galois representation are the canonical pair $\{(\ell+1)/4,(3\ell-1)/4\}$. The paper establishes non-surjective trace behavior for totally balanced curves, but notes that ordinary balancedness may not suffice to obtain analogous behavior or to complete the boundedness argument for the family of heavenly curves over cubic fields.

A uniform bound forcing total balancedness would allow the comparison with Bourdons bound for CM elliptic curves to be applied more broadly and could help establish boundedness of the relevant pairs $(\ell,K)$.