Let E1 be an elliptic curve over k given by V2= U3+ bU + c for scalars b and c and let C4 be a genus 2 curve which is a degree 4 cover of E1. Then there exists a scalar s such that a representation for C4 is given by Y2= F (X) where:
F (X) = s 3+ bs + c 27cs3− 18b2s2− 27bcs − 2b3− 27c2 (4b3+ 27c2)3(3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X 6 + 3 s 3+ bs + c2 3s2+ b (4b3+ 27c2)2(3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X 5 + 3 s 3+ bs + c E 4 (4b3+ 27c2)2(3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X 4 + − s 3+ bs + c2 G 2 (4b3+ 27c2) (3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X 3 + − s 3+ bs + c H 16 (4b3+ 27c2) (3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X 2 + 3 s 3+ bs + c2 3s4+ 6bs2+ 12cs − b2 J 16 (3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3X + − s 3+ bs + c JK 64 (3bs4+ 18cs3− 6b2s2− 6bcs − b3− 9c2)3 (B.2.1) and where E = 9cs7− 26b2s6− 171bcs5+ 34b3s4− 333c2s4+ 155b2cs3− 6b4s2+ 126bc2s2 + 7b3cs + 144c3s − 2b5− 17b2c2 G = 7s6+ 23bs4+ 68cs3− 11b2s2− 4bcs − 3b3− 20c2 H = 27cs11+ 6b2s10+ 585bcs9− 402b3s8+ 2349c2s8− 3330b2cs7+ 460b4s6 − 6156bc2s6+ 1410b3cs5− 7776c3s5+ 140b5s4+ 4230b2c2s4+ 23b4cs3 + 3024bc3s3+ 46b6s2+ 516b3c2s2+ 3024c4s2+ 5b5cs − 48b2c3s + 6b7 + 85b4c2+ 288bc4 J = s6+ 5bs4+ 20cs3− 5b2s2− 4bcs − b3− 8c2 K = 27cs9− 54b2s8− 324bcs7+ 36b3s6− 891c2s6+ 378b2cs5− 72b4s4 + 81bc2s4− 36b3cs3+ 324c3s3− 36b5s2− 297b2c2s2− 45b4cs − 324bc3s − 2b6− 45b3c2− 216c4.
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