In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by , with the maximum possible number of 45 nodes.
The equations defining the Burkhardt quartic become simpler if it is embedded in P<sup>5</sup> rather than P<sup>4</sup>. In this case it can be defined by the equations ÃÂ<sub>1</sub> = ÃÂ<sub>4</sub> = 0, where ÃÂ<sub>i</sub> is the ith elementary symmetric function of the coordinates (x<sub>0</sub> : x<sub>1</sub> : x<sub>2</sub> : x<sub>3</sub> : x<sub>4</sub> : x<sub>5</sub>) of P<sup>5</sup>.
The automorphism group of the Burkhardt quartic is the Burkhardt group U<sub>4</sub>(2) = PSp<sub>4</sub>(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group of E6.
The Burkhardt quartic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A<sub>2</sub>(3).