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Galois lines for space elliptic curve with j=12^3

The V_4-lines for each linearly normal space elliptic curve form the edges of a tetrahedron, however in case the elliptic curve has j=12^3, there exist Z_4-lines in addition. We show the arrangement of V_4 and Z_4-lines explicitly for the curve. As a corollary we obtain that each irreducible quartic curve with genus one has at most two Galois points.