Symplectic Linear Groups with GAP#

class sage.groups.matrix_gps.symplectic_gap.SymplecticMatrixGroup_gap(degree, base_ring, special, sage_name, latex_string, gap_command_string, category=None)[source]#

Bases: SymplecticMatrixGroup_generic, NamedMatrixGroup_gap, FinitelyGeneratedMatrixGroup_gap

Symplectic group in GAP.

EXAMPLES:

Sage

sage: Sp(2,4)                                                                   # needs sage.rings.finite_rings
Symplectic Group of degree 2 over Finite Field in a of size 2^2

sage: latex(Sp(4,5))
\text{Sp}_{4}(\Bold{F}_{5})

Python

>>> from sage.all import *
>>> Sp(Integer(2),Integer(4))                                                                   # needs sage.rings.finite_rings
Symplectic Group of degree 2 over Finite Field in a of size 2^2

>>> latex(Sp(Integer(4),Integer(5)))
\text{Sp}_{4}(\Bold{F}_{5})

invariant_form()[source]#

Return the quadratic form preserved by the symplectic group.

OUTPUT: a matrix

EXAMPLES:

Sage

sage: Sp(4, GF(3)).invariant_form()
[0 0 0 1]
[0 0 1 0]
[0 2 0 0]
[2 0 0 0]

Python

>>> from sage.all import *
>>> Sp(Integer(4), GF(Integer(3))).invariant_form()
[0 0 0 1]
[0 0 1 0]
[0 2 0 0]
[2 0 0 0]