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§4. Isomorphism Classification in Characteristics = 2, 3 73

4. Calculate the discriminant for the following elliptic curves. Calculate the value of j.
2
2
2
3
3
2
(a) y + y = x − x . (b) y + y = x + x .
2
3
2
3
(c) y + y = x − x. (d) y + y = x + x.
2 3 2 3
(e) y − xy + y = x . (f) y − y = x − 7.
3 2
5. Give the general formula for the discriminant for x + ax + bx and compare with the
2
3
discriminant of the cubic curve with equation y 2 = x + ax + bx. Determine the
discriminant of the following elliptic curves.
2 3 2 2 3 2
(a) y = x + x − x. (b) y = x − x + x.
2 3 2 3 2
(c) y = x − 2x − x. (d) y = x − 2x − 15x.
§4. Isomorphism Classification in Characteristics = 2, 3
In §4, §5, §6 we have a common approach to the isomorphism classification, and in
§8 there is an alternative approach to §4.
Let k be a field of characteristic = 2, 3 in this section. For an elliptic curve E over
k we can choose coordinates x and y giving the Weierstrass model in the following
form:
dx
2 3
y = x + a 4 x + a 6 and ω = .
2y
By (3.4) we see that
2
3
c 4 =−48a 4 , c 6 =−864a 6 , and =−16(4a + 27a ).
4 6
2
3
(4.1) Conditions for Smoothness. For f = y − x − a 4 x − a 6 the curve E is
smooth or nonsingular if and only if f, f x ,and f y have no common zero. By (3.6)
the curve E is smooth if and only if = 0.
Further, j = j(E) is given by
3
3
c 3 c 3 −4 12 (a 4 ) 3 4a 3
j = 4 = 12 3 4 = = 12 3 4 .
2
3
3
3
c − c 2 −16(4a + 27a ) 4a + 27a 2
4 6 4 6 4 6
(4.2) Isomorphisms Between Two Curves with the Same j-Invariant. Suppose E
2
3
¯
and E are two elliptic curves defined over k with equations y = x + a 4 x + a 6 and
3
2
y = x +¯a 4 x +¯a 6 such that j = j(E) = j(E).If φ : E → E is an isomorphism,
¯
¯
or equivalently admissible change of variables, then
2 3 4 6
xφ = u ¯x, yφ = u ¯y, a 4 = u ¯a 4 , and a 6 = u ¯a 6 .
3
These relations are now studied in three separate cases. Observe that 12 = 0inthe
characteristics under consideration.