Abstract
The 27 lines on a smooth cubic surface over \({{\mathbb Q}}\) are acted upon by a finite quotient of \({{\rm Gal}(\overline{\mathbb Q}/{\mathbb Q})}\) . We construct explicit examples such that the operation is via the index two subgroup of the maximal possible group. This is the simple group of order 25,920. Our examples are given in pentahedral normal form with rational coefficients. For such cubic surfaces, we study the discriminant and show its relation to the index two subgroup.
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Elsenhans, AS., Jahnel, J. The discriminant of a cubic surface. Geom Dedicata 159, 29–40 (2012). https://doi.org/10.1007/s10711-011-9643-7
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DOI: https://doi.org/10.1007/s10711-011-9643-7