Detailed magnetic field structure of the dense core SL42 (CrA-E) in the Corona Australis molecular cloud complex was investigated based on near-infrared polarimetric observations of background stars to measure dichroically polarized light produced by magnetically aligned dust grains. The magnetic fields in and around SL42 were mapped using 206 stars and curved magnetic fields were identified. On the basis of simple hourglass (parabolic) magnetic field modeling, the magnetic axis of the core on the plane of sky was estimated to be $40^{circ} pm 3^{circ}$. The plane-of-sky magnetic field strength of SL42 was found to be $22.4 pm 13.9$ $mu$G. Taking into account the effects of thermal/turbulent pressure and the plane-of-sky magnetic field component, the critical mass of SL42 was obtained to be $M_{rm cr} = 21.2 pm 6.6$ M$_{odot}$, which is close to the observed core mass of $M_{rm core} approx 20$ M$_{odot}$. We thus conclude that SL42 is in a condition close to the critical state if the magnetic fields lie near the plane of the sky. Since there is a very low luminosity object (VeLLO) toward the center of SL42, it is unlikely this core is in a highly subcritical condition (i.e., magnetic inclination angle significantly deviated from the plane of sky). The core probably started to collapse from a nearly kinematically critical state. In addition to the hourglass magnetic field modeling, the Inoue & Fukui (2013) mechanism may explain the origin of the curved magnetic fields in the SL42 region.
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