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As mentioned earlier, using <math>Y</math> and <math>y_{\bar{t}}^{q}</math> we are now able to determine which of the two values of <math>\bar{t}</math> (<math>\bar{t}</math> or <math>-\bar{t}</math>) works. This gives the value of <math>t\equiv \bar{t}\pmod l</math>. Schoof's algorithm stores the values of <math>\bar{t}\pmod l</math> in a variable <math>t_l</math> for each prime <math>l</math> considered.
===Case 2 <math>(x^{q^{2}}, y^{q^{2}}) = \pm \bar{q}(x, y)</math>===
We begin with the assumption that <math>(x^{q^{2}}, y^{q^{2}}) = \bar{q}(x, y)</math>. Since <math>l</math> is an odd prime it cannot be that <math>\bar{q}(x, y)=-\bar{q}(x, y)</math> and thus <math>\bar{t}\neq 0</math>. The characteristic equation yields that <math>\bar{t} \phi(P) = 2\bar{q} P</math>. And consequently that <math>\bar{t}^{2}\bar{q} \equiv (2q)^{2} \pmod l</math>.
This implies that <math>q</math> is a square modulo <math>l</math>. Let <math>q \equiv w^{2} \pmod l</math>. Compute <math>w\phi(x,y)</math> in <math>\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \psi_{l})</math> and check whether <math>
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