ellisomat

Function: ellisomat
Section: elliptic_curves
C-Name: ellisomat
Prototype: GD0,L,D0,L,
Help: ellisomat(E, {p=0}, {fl=0}): E being an elliptic curve over a number
 field K, return a list of representatives of the isomorphism classes of
 elliptic curves defined over K and K-isogenous to E, with the corresponding
 isogenies from E and their dual, and the matrix of the isogeny degrees between
 the curves. If the flag fl is 1, the isogenies are not computed, which saves
 time. If p is set, it must be a prime number: in this case only isogenies of
 degree a power of p are considered.
Doc:
 Given an elliptic curve $E$ defined over a number field $K$, compute
 representatives of the isomorphism classes of elliptic curves defined over
 $K$ and $K$-isogenous to $E$. We assume that $E$ does not have CM over $K$
 (otherwise that set would be infinite).
 For any such curve $E_i$, let $f_i: E \to E_i$ be a rational isogeny
 of minimal degree and let $g_i: E_i \to E$ be the dual isogeny; and let $M$
 be the matrix such that $M_{i,j}$ is the minimal degree for an isogeny $E_i
 \to E_j$.

 The function returns a vector $[L,M]$ where $L$ is a list of triples
 $[E_i, f_i, g_i]$ ($\fl  = 0$), or simply the list of $E_i$ ($\fl = 1$,
 which saves time). The curves $E_i$ are given in $[a_4,a_6]$ form and the
 first curve $E_1$ is isomorphic to $E$ by $f_1$.

 If $p$ is set, it must be a prime number; in this which case only isogenies of
 degree a power of $p$ are considered.

 Over a number field, the possible isogeny degrees are determined by
 Billerey algorithm.
 \bprog
 ? E = ellinit("14a1");
 ? [L,M] = ellisomat(E);
 ? LE = apply(x->x[1], L)  \\ list of curves
 %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864],
      [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]]
 ? L[2][2]  \\ isogeny f_2
 %4 = [x^3+3/4*x^2+19/2*x-311/12,
       1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3]
 ? L[2][3]  \\ dual isogeny g_2
 %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64,
       -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x
       +(-3601/64*y+16947/512),x-3/4]
 ? apply(E->ellidentify(ellinit(E))[1][1], LE)
 %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"]
 ? M
 %7 =
 [1  3  3 2  6  6]

 [3  1  9 6  2 18]

 [3  9  1 6 18  2]

 [2  6  6 1  3  3]

 [6  2 18 3  1  9]

 [6 18  2 3  9  1]
 @eprog