$2i$ map of an the elliptic curve over $F_p^2$ defined by $y^2=x^3+x$

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sage: F = GF(127^2)
sage: E = EllipticCurve(F, [1,0])
sage: i = E.automorphisms()[-1]
sage: assert i^2 == -1
sage: endo = 2 * i
sage: assert endo in End(E)
sage: assert endo^2 == -4
sage: endo
Composite morphism of degree 4 = 1*4:
  From: Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 127^2
  To:   Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 127^2