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Quadrupole transitions of $^{10}$C and their isospin symmetry with $^{10}$Be

We investigate the structures of $^{10}$C focusing on the quadrupole properties in comparison with the mirror nucleus $^{10}$Be. We describe $^{10}$C and $^{10}$Be in the variation of the multiple bases of the antisymmetrized molecular dynamics (AMD), in which the multiple AMD bases are optimized simultaneously in the total-energy variation. In the monopole transitions, we confirm the isospin symmetry between $^{10}$C and $^{10}$Be by exchanging protons and neutrons. In the quadrupole transitions, most cases show larger values in $^{10}$C than those of $^{10}$Be, except for the transition of $2^+_1\to 0^+_1$. The transition of $2^+_1\to 0^+_1$ shows similar values in the two nuclei in spite of the different proton numbers, which agrees with the experimental situation as an anomaly. This relation comes from the small proton deformation in $^{10}$C due to its subclosed nature and the large proton deformation in $^{10}$Be due to two-$α$ clustering. This property can also be seen in the quadrupole moments of the two nuclei. In the neutron deformations of $^{10}$C and $^{10}$Be, the opposite tendency of protons is confirmed and these results ensure the isospin symmetry between the two nuclei. We also confirm the large quadrupole transitions between the elongated linear-chain states. It would be desirable for future experiments to investigate the present characteristics of the transitions in the two nuclei.