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.
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