We use direct numerical simulations to study homogeneous and isotropic turbulent flows of dilute polymer solutions at high Reynolds and Deborah numbers. We find that for small wave numbers k, the kinetic energy spectrum shows Kolmogorov-like behavior that crosses over at a larger k to a novel, elastic scaling regime, E(k) similar to k-xi, with xi approximate to 2.3. We study the contribution of the polymers to the flux of kinetic energy through scales and find that it can be decomposed into two parts: one increase in effective viscous dissipation and a purely elastic contribution that dominates over the nonlinear flux in the range of k over which the elastic scaling is observed. The multiscale balance between the two fluxes determines the crossover wave number that depends nonmonotically on the Deborah number. Consistently, structure functions also show two scaling ranges, with intermittency present in both of them in equal measure.