For a 1-parametric family \(\mathcal {E}_{U}\) of elliptic curves over \(\mathbb {Q}\) and a prime p, consider the second moment sum \(M_{2,p}(\mathcal {E}_U)=\sum _{u\in \mathbb {F}_{p}}a_{u,p}^2\) , where \(a_{u,p}=p+1-\#\mathcal {E}_{u}(\mathbb {F}_{p})\) . Inspired by Rosen and Silverman’s proof of the Nagao conjecture, which relates the first moment of a rational elliptic surface to the rank of the Mordell–Weil group of the corresponding elliptic curve, S. J. Miller initiated the study of the asymptotic expansion of \(M_{2,p}(\mathcal {E}_U)=p^2+O(p^{3/2})\) (which by the work of Deligne and Michel has a cohomological interpretation). He conjectured that, similar to the first moment case, the largest lower-order term that does not average to 0 has a negative bias. In this paper, we provide an explicit formula for the second moment \(M_{2,p}(\mathcal {E}_{U})\) of \(\begin{aligned} \mathcal {E}_{U}:y^2=P(x)U+Q(x), \end{aligned}\) where \(\deg P(x),\deg Q(x)\le 3\) . For a generic choice of polynomials P(x) and Q(x) this formula is expressed in terms of the point count of a certain genus two curve. As an application, we prove that the Bias conjecture holds for the pencil of the cubics \(\mathcal {E}_U\) .
