Algorithm for computing ` kernel polynomials

mials

LetEbe an elliptic curve over a fieldK and letψ` be the`-division polynomial ofE. Here we summarize the method described in this chapter for constructing

K-rational `-kernel polynomials of E by factorizingψ`.

Remark 3.5.1. For this algorithm, it is best to take the smallest semi- primitive root modulo ` to reduce the computation expense. As ` becomes large, computation becomes very expensive since we need to factorizeψ`which has degree `2₂−1. In subsequent chapters, we will develop methods which only require factorization of polynomials of degree `+ 1.

Algorithm 1 Computing K-rational `-kernel polynomials by factoring the

`-division polynomial.

Input: E, K, `, a (a semi-primitive root modulo `) Output: A set of K-rational `-kernel polynomials of E

1: Compute ψ`

2: Factorize ψ` over K

3: if `= 2 or 3 then

4: Set K={linear factors of ψ` overK}

5: else

6: Set F ={irreducible factors ofψ` overK of degree dividing `−₂1}

7: Set K=∅

8: for f inF do

9: Letd= deg(f) ande= `₂−_d1

10: Compute f, τa(f), ..., τae(f) 11: if τe a(f) =f then 12: Append Qe_i=0−1τi a(f) to K 13: end if 14: Deleteτi a(f) from F for 0≤i≤e−1 15: end for 16: end if 17: return K

We give the following two examples that use Algorithm 1. Example 3.5.2. Let E be the elliptic curve over _Q(√5) given by

E : y2+y=x3−x2−10x−20.

We will compute all _Q(√5)-rational 5-kernel polynomials of E. Note that a 5-kernel polynomial ofE has degree 5−₂1 = 2. The 5-division polynomialψ5 of

E factorizes as ψ5 =Q8_i=1fi where: f1 =x−16, f2 =x−5, f3 =x+ 1₂ − 11 √ 5 10 , f4 =f3, f5 =x2+ 15−5 √ 5 2 x+ 95−41√5 2 , f8 =f5, f6 =x2+ 1−3 √ 5 2 x+ 11 + 2 √ 5, f7 =f6,

root modulo 5. Sinceτ2(f1) =f2 and τ22(f1) = f1, we obtain that

f1·f2 = (x−16)(x−5) =x2−21x+ 80

is a 5-kernel polynomial of E. Moreover, since τ2(f3) = f4 and τ22(f3) = f3,

the polynomial

f3·f4 =x2+x−

29 5 is also a 5-kernel polynomial ofE.

However, the polynomialsf5,f6,f7 andf8 are not 5-kernel polynomials

of E, since they do not satisfy τ2(fi) = fi even though they are irreducible quadratic factors;τ2 permutes f5,f6 and permutesf7,f8. Hence there are two

Q(

√

5)-rational 5-kernel polynomials ofE, namelyx2−21x+80 andx2+x−29 5 ,

which are in fact defined over_Q.

Example 3.5.3. This is a revisit of Example 3.1.1. LetEbe the elliptic curve over_F3 given by

E : y2 =x3−x.

We will compute all _F3-rational 13-kernel polynomials of E. Note that a 13-

kernel polynomial of E has degree 13₂−1 = 6. The 13-division polynomial ψ13

of E factorizes as ψ13 = Q14 i=1fi where: f1 =x6 +x4+x3+x2−x−1, f2 =x6+x4 −x3 +x2+x−1, f3 =x6 +x5+x3−x2+x−1, f4 =x6+x5 −x3 −x2 −1, f5 =x6 +x5+x4+x2−x−1, f6 =x6+x5 +x4+x3+x2+x−1, f7 =x6 +x5−x4+x3+x−1, f8 =x6+x5 −x4 −x3 −1, f9 =x6 −x5 +x3−x2−1, f10 =x6−x5−x3−x2−x−1, f11=x6−x5+x4+x2+x−1, f12 =x6−x5+x4−x3+x2−x−1, f13=x6−x5−x4+x3 −1, f14 =x6−x5−x4−x3−x−1.

Note that 2 is a semi-primitive root modulo 13. Since eachfi has degree 6, fi is a 13-kernel polynomial if and only if τ2(fi) = fi for each i= 1, ...,14. One can check that

τ2(f1) =f1 and τ2(f2) =f2,

so f1 and f2 are 13-kernel polynomials of E. However, for other factors we

τ2(f3) = f14, τ2(f4) =f9, τ2(f5) =f11,

τ2(f6) = f13, τ2(f7) =f10, τ2(f8) =f12,

thus fi are not 13-kernel polynomials E for i = 3, ...,14. Hence E has two

F3-rational 13-kernel polynomials, namelyf1 =x6+x4+x3+x2−x−1 and

f2 =x6+x4−x3+x2+x−1.

Now we see what happens over _F9. Over F9, ψ13 factorizes as ψ13 =

Q28

i=1hi where hi are as follows and a denotes an element in F9 such that

a2 =−1: h1 =x3 −x+a+ 1, h3 =h1, h2 =x3 −x+a−1, h4 =h2, h5 =x3 + (a+ 1)x2 +a−1, h17=h5, h6 =x3 + (a+ 1)x2 +x+a−1, h18=h6, h7 =x3 + (a+ 1)x2 + (a+ 1)x−a−1, h21=h7, h8 =x3 + (a+ 1)x2 + (a−1)x−a+ 1, h22=h8, h9 =x3 + (a+ 1)x2 −ax−a−1, h19=h9, h10=x3+ (a+ 1)x2+ (−a−1)x−a+ 1, h20=h10, h11=x3+ (a−1)x2+a+ 1, h23=h11, h12=x3+ (a−1)x2+x+a+ 1, h24=h12, h13=x3+ (a−1)x2+ax−a+ 1, h27=h13, h14=x3+ (a−1)x2+ (a−1)x−a−1, h28=h14, h15=x3+ (a−1)x2+ (−a+ 1)x−a+ 1, h25=h15, h16=x3+ (a−1)x2+ (−a−1)x−a−1, h26=h16,

wherehi denotes the conjugate of hi with respect to the conjugationa 7→ −a. By using τ2, we obtain the following 14 F9-rational 13-kernel polynomials of

E:

h1·τ2(h1) = h1·h3 =x6+x4−x3+x2+x−1,

h2·τ2(h2) = h2·h4 =x6+x4+x3+x2−x−1,

h5·τ2(h5) = h5·h25=x6+ (−a+ 1)x4+ax3+ (−a−1)x2+x+ 1,

h9·τ2(h9) = h9·h16=x6−ax5+ax4−x3+ (−a+ 1)x2−x−a, h10·τ2(h10) = h10·h28 =x6+ (−a+ 1)x4+ (−a−1)x2 −a, h11·τ2(h11) = h11·h21 =x6+ (a+ 1)x4+ax3+ (a−1)x2 −x+ 1, h14·τ2(h20) = h14·h20 =x6+ (a+ 1)x4+ (a−1)x2+a, h15·τ2(h15) = h15·h17 =x6+ (a+ 1)x4−ax3+ (a−1)x2+x+ 1, h18·τ2(h18) = h18·h24 =x6+ax5−ax3 +x2+ax+ 1, h19·τ2(h19) = h19·h26 =x6+ax5−ax4 −x3 + (a+ 1)x2 −x+a, h22·τ2(h22) = h22·h27 =x6+ax5+ax4+x3+ (−a+ 1)x2+x−a.

Chapter 4

Modular Approach: cases where

X

₀

(`)

has genus

0

Recall from section 2.3.2 that Ell0_0,N(K) denotes the set of pairs (E, H), up to isomorphism over K, where E is an elliptic curve defined over K and

H ⊆ E(K) is a cyclic subgroup of order N that is defined over K. Then there is a bijection Ell0_0,N(K)−→∼ X0(N)Q(K)\ {cusps}. Using this moduli in-

terpretation, we compute`-isogenies of an elliptic curve for` ∈ {2,3,5,7,13}, following Cremona and Watkins [3]. In this chapter except for section 4.5, we assume that char(K) 6∈ {2,3, `}. In section 4.5 we consider the cases where char(K) = 2 or 3.