72 2. Elliptic curves

(2) Show that if y2 = f(x) is singular, where f(x) ∈ K[x] is a

monic cubic polynomial, then the singularity must occur at

(δ,0), where δ is a root of f(x).

(3) Show that (δ,0) is singular if and only if δ is a double root

of f(x). Therefore D = 0 if and only if E is singular.

Exercise 2.12.10. Let E/Q : y2 = x3 + 3. Find all the points of

E(F7) and verify that N7 satisfies Hasse’s bound.

Exercise 2.12.11. Let E/Q : y2 = x3 + Ax + B and let p ≥ 3 be

a prime of bad reduction for E/Q. Show that E(Fp) has a unique

singular point.

Exercise 2.12.12. Prove parts (1) and (3) of Theorem 2.8.5. (Hint:

use Definition 2.8.4 and Proposition 2.7.3.)

Exercise 2.12.13. Prove Corollary 2.8.6.

Exercise 2.12.14. Let E :

y2

=

x3

− 10081x. Use Sage (or PARI)

to find a minimal set of generators for the subgroup that is spanned

by all these points on E:

(0,0), (−100,90),

10081

100

,

90729

1000

, (−17,408)

907137

6889

, −

559000596

571787

,

1681

16

,

20295

64

,

833

4

,

21063

8

−

161296

1681

,

19960380

68921

, −

6790208

168921

, −

40498852616

69426531

.

(Hint: use Theorem 2.7.4 to determine the rank of E/Q.)

Exercise 2.12.15. Let E and δ be defined as in Theorem 2.9.3, and

suppose P = (x0,y0) is a point on E with y0 = 0. Show:

• δ(P ) · δ(O) = δ(P ).

• δ((e1, 0)) · δ((e2, 0)) = δ((e1, 0) + (e2,0)).

• δ(P ) · δ((e1, 0)) = δ(P + (e1,0)).

Exercise 2.12.16. Let E :

y2

=

x3

+Ax+B be an elliptic curve with

A, B ∈ Q, and suppose P = (x0,y0) is a point on E, with y0 = 0.