By using isochronous mass spectrometry (IMS) at the experimental cooler storage ring CSRe, masses of short-lived $^{44}$Cr, $^{46}$Mn, $^{48}$Fe, $^{50}$Co and $^{52}$Ni were measured for the first time and the precision of the mass of $^{40}$Ti was improved by a factor of about 2. Relative precisions of $delta m/m=(1-2)times$10$^{-6}$ have been achieved. Details of the measurements and data analysis are described. The obtained masses are compared with the Atomic-Mass Evaluation 2016 (AME$^{prime}$16) and with theoretical model predictions. The new mass data enable us to extract the higher order coefficients, $d$ and $e$, of the quartic form of the isobaric multiplet mass equation (IMME) for the $fp$-shell isospin quintets. Unexpectedly large $d$- and $e$-values for $A=44$ quintet are found. By re-visiting the previous experimental data on $beta$-delayed protons from $^{44}$Cr decay, it is suggested that the observed anomaly could be due to the misidentification of the $T=2$, $J^pi=0^{+}$ isobaric analog state (IAS) in $^{44}$V.
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