We investigate sc7(n), the number of self-conjugate 7-core partitions of size n. It turns out that sc7(n)=0 for n≡7(mod8). For n≡1,3,5(mod8), with n≢5(mod7), we find that sc7(n) is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show thatsc7(n)=2−ε(n)−1⋅H(−Dn), where −Dn:=−4ε(n)(7n+14) and ε(n):=12⋅(1+(−1)n−12). This fact implies several corollaries which are of interest. For example, if −Dn is a fundamental discriminant and p∉{2,7} is a prime with ordp(−Dn)≤1, then for every positive integer k we have(1)sc7((n+2)p2k−2)=sc7(n)⋅(1+pk+1−pp−1−pk−1p−1.(−Dnp)), where (−Dnp) is the Legendre symbol.