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

TL;DR

This paper investigates the structure of Galois lines for space elliptic curves with j-invariant 12^3, revealing additional Z_4-lines and their arrangements, and deriving bounds on Galois points for genus one quartic curves.

Contribution

It explicitly describes the arrangement of V_4 and Z_4-lines for elliptic curves with j=12^3, highlighting new Z_4-line existence and bounds on Galois points.

Findings

V_4-lines form a tetrahedron for general space elliptic curves

Existence of Z_4-lines for j=12^3 elliptic curves

Maximum of two Galois points for genus one quartic curves

Abstract

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.