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## Example: Computing the Matrix of Frobenius for at

Let and consider the elliptic curve , with minimal model .

Step 1.
Put the elliptic curve into Weierstrass form , via the transformation

In our case, we obtain the curve

Let

Step 2.
Fix the precision and compute . In our case, and .
Step 3.
Compute the action of Frobenius on the two differentials and as an element of , with a precision of digits. Furthermore, group the terms of as , where the are in of degree less than 3.

In our case, we compute

Step 4.
Now we must reduce the differentials. We want to write each of the

as

in . We begin with

and compute the appropriate list of differentials:
 0 1 2 0 1 2

Thus we wish to write as a linear combination of , , and , all modulo 25 (we may ignore the lower powers of present in the differentials, as we will take care of them in the steps to come). We find that taking

leaves us with

Now we wish to write as a linear combination of , , and , modulo 25. We find that taking

leaves us with

Next, we reduce

Note that this has an term, so we take care of this first:

Now we proceed as in the case of , and we wish to write as a linear combination of , , and , all modulo 25. We find that taking

leaves us with

Finally, we wish to write as a linear combination of , , and , all modulo 25. We find that taking

leaves us with

Step 5.
Now we form the matrix of the reduced differentials, where each reduced differential gives us a column in the matrix of absolute Frobenius. In our case, we have .
As a consistency check, we have that has trace 23, which is modulo 25 and determinant , which is modulo 25.

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William Stein 2006-10-20