A computational approach to L\"uroth quartics

A plane quartic curve is called L\"uroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic 4-fold, associated to any plane quartic. We review their construction and we show how it gives a computational tool to detect if a plane quartic is L\"uroth. As a byproduct, the 28 bitangents of a general plane quartic correspond to 28 singular points of the associated White-Miller quartic 4-fold.