  The moments of the coefficients of elliptic curve L-functions are related to
numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao
relating the first moment of one-parameter families satisfying Tate's
conjecture to the rank of the corresponding elliptic surface over Q(T); one can
also construct families of moderate rank by finding families with large first
moments. Michel proved that if j(T) is not constant, then the second moment of
the family is of size p^2 + O(p^(3/2)); these two moments show that for
suitably small support the behavior of zeros near the central point agree with
that of eigenvalues from random matrix ensembles, with the higher moments
impacting the rate of convergence.
  In his thesis, Miller noticed a negative bias in the second moment of every
one-parameter family of elliptic curves over the rationals whose second moment
had a calculable closed-form expression, specifically the first lower order
term which does not average to zero is on average negative. This Bias
Conjecture is confirmed for many families; however, these are highly
non-generic families whose resulting Legendre sums can be determined. Inspired
by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey
Pozdnyakov and others in investigations of murmurations of elliptic curve
coefficients with machine learning techniques, we pose a similar problem for
trying to understand the Bias Conjecture. As a start to this program, we
numerically investigate the Bias Conjecture for a family whose bias is positive
for half the primes. Since the numerics do not offer conclusive evidence that
negative bias for the other half is enough to overwhelm the positive bias, the
Bias Conjecture cannot be verified for the family.
