  In the cuprate superconductors, Nernst and torque magnetization experiments
have provided evidence that the disappearance of the Meissner effect at $T_c$
is caused by the loss of long-range phase coherence, rather than the vanishing
of the pair condensate. Here we report a series of torque magnetization
measurements on single crystals of $\mathrm{La_{2-x}Sr_xCuO_4}$ (LSCO),
$\mathrm{Bi_2Sr_{2-y}La_yCuO_6}$ (Bi 2201),
$\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ (Bi 2212) and optimal
$\mathrm{YBa_2Cu_3O_7}$. Some of the measurements were taken to fields as high
as 45 T. Focusing on the magnetization above $T_c$, we show that the
diamagnetic term $M_d$ appears at an onset temperature $T^M_{onset}$ high above
$T_c$. We construct the phase diagram of both LSCO and Bi 2201 and show that
$T^M_{onset}$ agrees with the onset temperature of the vortex Nernst signal
$T^{\nu}_{onset}$. Our results provide thermodynamic evidence against a recent
proposal that the high-temperature Nernst signal in LSCO arises from a
quasiparticle contribution in a charge-ordered state.
