Planes of the Burkhardt quartic

This page displays equations of the 40 planes in the Burkhardt quartic X₄ defined in P⁴ by

    y[1]^4-y[1]*(y[2]^3+y[3]^3+y[4]^3+y[5]^3)+3*y[2]*y[3]*y[4]*y[5].

Here is the source code.


The  1-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[5]
------------------------


The  2-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[2],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[2]
------------------------


The  3-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5]
------------------------


The  4-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - y[3] - y[4] - zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[2]
------------------------


The  5-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The  6-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - 
    zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[2]
------------------------


The  7-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The  8-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - 
    zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[2]
------------------------


The  9-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[2],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5]
------------------------


The 10-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - y[3] - y[4] - zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[5]
------------------------


The 11-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[2],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 12-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - 
    zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[5]
------------------------


The 13-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (-2*zeta_3 - 1)*y[2],
zeta_3*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 14-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - 
    zeta_3*y[5],
(zeta_3 + 2)*y[1] + (2*zeta_3 + 1)*y[5]
------------------------


The 15-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - y[3] - y[4] - zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 16-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - 
    zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5]
------------------------


The 17-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - y[3] - y[4] - zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 18-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - 
    zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5]
------------------------


The 19-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - 
    zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 20-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - 
    zeta_3*y[5],
zeta_3*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 
    1)*y[5]
------------------------


The 21-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (zeta_3 - 1)*y[5],
zeta_3*y[1] - y[2] - y[3] - y[4] - zeta_3*y[5]
------------------------


The 22-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] - y[5],
(zeta_3 + 2)*y[1] + (-zeta_3 - 2)*y[2]
------------------------


The 23-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (zeta_3 - 1)*y[5],
zeta_3*y[1] - y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - zeta_3*y[5]
------------------------


The 24-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - y[5],
(zeta_3 + 2)*y[1] + (-zeta_3 - 2)*y[2]
------------------------


The 25-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 + 1)*y[1] + (zeta_3 - 1)*y[5],
zeta_3*y[1] - y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - zeta_3*y[5]
------------------------


The 26-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - y[5],
(zeta_3 + 2)*y[1] + (-zeta_3 - 2)*y[2]
------------------------


The 27-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] - y[5],
zeta_3*y[1] - y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - zeta_3*y[5]
------------------------


The 28-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - y[5],
zeta_3*y[1] - y[2] - y[3] - y[4] - zeta_3*y[5]
------------------------


The 29-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - y[3] - y[4] - y[5],
zeta_3*y[1] - y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - zeta_3*y[5]
------------------------


The 30-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - y[5],
zeta_3*y[1] - y[2] - y[3] - y[4] - zeta_3*y[5]
------------------------


The 31-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - y[5],
zeta_3*y[1] - y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - zeta_3*y[5]
------------------------


The 32-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] + (zeta_3 + 1)*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - y[5],
zeta_3*y[1] - y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - zeta_3*y[5]
------------------------


The 33-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - y[5]
------------------------


The 34-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] - y[3] - y[4] - y[5]
------------------------


The 35-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] - y[3] - y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - y[5]
------------------------


The 36-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] - y[3] - y[4] - y[5]
------------------------


The 37-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] - y[5]
------------------------


The 38-th plane in X4 is given by:
Scheme over K defined by
(-zeta_3 - 1)*y[1] - y[2] + (zeta_3 + 1)*y[3] - zeta_3*y[4] + (zeta_3 + 1)*y[5],
zeta_3*y[1] - zeta_3*y[2] - zeta_3*y[3] + (zeta_3 + 1)*y[4] - y[5]
------------------------


The 39-th plane in X4 is given by:
Scheme over K defined by
(zeta_3 - 1)*y[1] + (2*zeta_3 + 1)*y[4],
(-zeta_3 - 2)*y[1] + (-2*zeta_3 - 1)*y[4]
------------------------


The 40-th plane in X4 is given by:
Scheme over K defined by
(zeta_3 - 1)*y[1] + (2*zeta_3 + 1)*y[3],
(-zeta_3 - 2)*y[1] + (-2*zeta_3 - 1)*y[3]
------------------------