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We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.

Explicit moduli spaces for congruences of elliptic curves   (20 pages)          

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There are two accompanying Magma files that check the calculations in the paper, available here and here.