Geometriae Dedicata

, Volume 159, Issue 1, pp 29–40 | Cite as

The discriminant of a cubic surface

  • Andreas-Stephan Elsenhans
  • Jörg JahnelEmail author
Original Paper


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.


Cubic surface Pentahedral normal form Discriminant Rational point 

Mathematics Subject Classification (2000)

14J45 14J20 11G35 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Département MathematikUniversität SiegenSiegenGermany

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