Flat torus circle packings and Groebner algorithm

We are given the following packing of the flat torus with 25 circles below : Our goal is to compute the minimal polynomial of the corresponding circle diameter $l$. Using classical trigonometric identities we obtain a polynomial system of equations which consists of 13 equations, 12 trigonometric variables (the sines and cosines of edges in the contact graph) + the diameter variable $l$. All approximate values of the solution are know up to 15 decimal places. The details of the system of equations and numerical values are available here. We tried to use a sage implementation of the Groebner Algorithm to eliminate trigonometric variables in order to obtain the characteristic polynomial for the diameter $l$. It has been running for days without producing any anwser. the implementation of our sage script is also available here. For anyone interested I also added the optimal solution for a packing with 21 circles here. Also available our sage script to compute the corresponding minimal polynomial here. After 17 minutes of computations it returns a polynomial of degree 136. Are there other tools we could use in order to perform this task? I am not really familiar with optimizing Groebner algorithm so any good idea would be very much appreciated. Thanks!