Computing the kernel of an isogeny between two elliptic curves

Question

Consider the two rational elliptic curves -

$ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $

$ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $

We know that there is an isogeny of degree 9 between the above two curves. My question is how to write down the isogeny and compute it's kernel $?$

EDIT - Is there any theoretical way of calculating the kernel since I don't know how to find out the kernel from the kernel polynomial $?$

Answer 1

You can use Magma. The code

E:=EllipticCurve("35a2");

E2:=IsogenousCurves(E)[1];

A,B:=IsIsogenous(E,E2);

A;

B;

returns

true

Elliptic curve isogeny from: CrvEll: E to CrvEll: E2 taking (x : y : 1) to ((1/81*x^9 + 2/9*x^8 + 13*x^7 + 3392/27*x^6 - 30325/9*x^5 - 85999*x^4 - 24206654/27*x^3 - 50989888/9*x^2 - 199148981/9*x - 3252102320/81) / (x^8 + 134/3*x^7 + 4687/9*x^6 - 7022/3*x^5 - 655945/9*x^4 - 6723838/27*x^3 + 17079682/9*x^2 + 41442544/3*x + 1927297801/81) : (1/729*x^12*y - 364/729*x^12 + 67/729*x^11*y - 24388/729*x^11 - 194/729*x^10*y - 557539/729*x^10 - 1985/243*x^9*y - 1017385/243*x^9 + 656695/729*x^8*y + 68692145/729*x^8 + 26025217/729*x^7*y + 1054678562/729*x^7 + 192550078/243*x^6*y + 544801208/243*x^6 + 8480552422/729*x^5*y - 57023892268/729*x^5 + 80725047070/729*x^4*y - 346491431755/729*x^4 + 475003324045/729*x^3*y + 525131058245/729*x^3 + 1557625275491/729*x^2*y + 11187520474201/729*x^2 + 2082152034392/729*x*y + 37883300781112/729*x - 467114355359/729*y + 1558207155673/27) / (x^12 + 67*x^11 + 4588/3*x^10 + 225865/27*x^9 - 187555*x^8 - 25720147/9*x^7 - 99672482/27*x^6 + 1512695518/9*x^5 + 28655848540/27*x^4 - 63917643605/81*x^3 - 771015395293/27*x^2 - 909684562072/9*x - 84610300761701/729) : 1)

and if you use now Kernel(B) then you get

Subgroup scheme of E defined by x^4 + 67/3*x^3 + 11*x^2 - 1416*x - 43901/9