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Let be a nonsingular plane quintic curve over the complex number field , and let be a projection from . Let be the Galois closure of the field extension induced by , where and are the rational function fields of and , respectively. We call the point a -point if the Galois group of is isomorphic to the dihedral group of order eight. In this paper, we prove that the number of -points for equals , , , , or , and show that the curve with -points is projectively equivalent to the Fermat quintic curve.
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Takeshi Takahashi, Projection of a nonsingular plane quintic curve and the dihedral group of order eight. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 39–61DOI 10.4171/RSMUP/135-3