Function: ellpadicfrobenius
Section: elliptic_curves
C-Name: ellpadicfrobenius
Prototype: GUL
Help: ellpadicfrobenius(E,p,n): matrix of the Frobenius at p>2 in the standard
 basis of H^1_dR(E) to absolute p-adic precision p^n.
Doc: If $p>2$ is a prime and $E$ is an elliptic curve on $\Q$ with good
 reduction at $p$, return the matrix of the Frobenius endomorphism $\varphi$
 on the crystalline module $D_{p}(E)= \Q_{p} \otimes H^{1}_{dR}(E/\Q)$ with
 respect to the basis of the given model $(\omega, \eta=x\*\omega)$, where
 $\omega = dx/(2\*y+a_{1}\*x+a_{3})$ is the invariant differential.
 The characteristic polynomial of $\varphi$ is $x^{2} - a_{p}\*x + p$.
 The matrix is computed to absolute $p$-adic precision $p^{n}$.

 \bprog
 ? E = ellinit([1,-1,1,0,0]);
 ? F = ellpadicfrobenius(E,5,3);
 ? lift(F)
 %3 =
 [120 29]

 [ 55  5]
 ? charpoly(F)
 %4 = x^2 + O(5^3)*x + (5 + O(5^3))
 ? ellap(E, 5)
 %5 = 0
 @eprog
