  We study the most luminous known supernova (SN) associated with a gamma-ray
burst (GRB), SN 2011kl. The photospheric velocity of SN 2011kl around peak
brightness is $21,000\pm7,000$ km s$^{-1}$. Owing to different assumptions
related to the light-curve (LC) evolution (broken or unbroken power-law
function) of the optical afterglow of GRB 111209A, different techniques for the
LC decomposition, and different methods (with or without a near-infrared
contribution), three groups derived three different bolometric LCs for SN
2011kl. Previous studies have shown that the LCs without an early-time excess
preferred a magnetar model, a magnetar+$^{56}$Ni model, or a white dwarf tidal
disruption event model rather than the radioactive heating model. On the other
hand, the LC shows an early-time excess and dip that cannot be reproduced by
the aforementioned models, and hence the blue-supergiant model was proposed to
explain it. Here we reinvestigate the energy sources powering SN 2011kl. We
find that the two LCs without the early-time excess of SN 2011kl can be
explained by the magnetar+$^{56}$Ni model, and the LC showing the early excess
can be explained by the magnetar+$^{56}$Ni model taking into account the
cooling emission from the shock-heated envelope of the SN progenitor,
demonstrating that this SN might primarily be powered by a nascent magnetar.
