Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 252

Exercises

8.1 Show that each of the following elliptic curves has the stated torsion
group.

(a) y 2 = x 3 2; 0

(b) y 2 = x 3 +8; Z 2
(c) y 2 = x 3 +4; Z 3
(d) y 2 = x 3 +4 x ; Z 4
(e) y 2 = x 3

−

− 432 x + 8208; Z 5
(f) y 2 = x 3 +1; Z 6
(g) y 2 = x 3 − 1323 x + 6395814; Z 7

(h) y 2 = x 3

− 44091 x + 3304854; Z 8

(i) y 2 = x 3 − 219 x + 1654; Z 9

(j) y 2 = x 3

− 58347 x + 3954150; Z 10

(k) y 2 = x 3

− 33339627 x + 73697852646; Z 12

(l) y 2 = x 3 − x ; Z 2 ⊕ Z 2

(m) y 2 = x 3

− 12987 x − 263466; Z 4 ⊕ Z 2

(n) y 2 = x 3

− 24003 x + 1296702; Z 6 ⊕ Z 2

(o) y 2 = x 3 − 1386747 x + 368636886; Z 8 ⊕ Z 2

Parameterizations of elliptic curves with given torsion groups can be
found in [67].

8.2 Let E be an elliptic curve over Q given by an equation of the form
y 2 = x 3 + Cx 2 + Ax + B ,with A, B, C ∈ Z .

(a) Modify the proof of Theorem 8.1 to obtain a homomorphism

λ r : E r /E 3 r −→ Z p 2 r

(see [68, pp. 51-52]).
(b) Show that ( x, y ) ∈ E ( Q ) is a torsion point, then x, y ∈ Z .

(a) Show that the map λ r , applied to the curve y 2 = x 3 ,isthemapof
Theorem 2.30 divided by p r and reduced mod p 4 r .
(b) Consider the map λ r of Exercise 8.2, applied to the curve E : y 2 =
x 3 + ax 2 .Let ψ be as in Theorem 2.31. The map λ r ψ − 1

8.3

gives a

map

y + αx
y − αx →

p −r x
y

(mod p 2 r ) .

Use the Taylor series for log((1 + t ) / (1 − t )) to show that the map
(2 α ) λ r ψ − 1 is p −r times the logarithm map, reduced mod p 2 r .