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§6. Isomorphism Classification in Characteristic 2 79

3
2
2
3
2
∼
E 4 : y + y = x + x + 1 = E : y + y = x + x + 1,

4
3
2
2

3 ∼
E 5 : y + y = x = E : y + y = x + 1,
5
2
3
2

∼
= E : y + y = x + x + x,
5
2
2
3
∼ 5 : y + y = x + x + x + 1.
= E
We can “graph” the elliptic curves E 3 , E 4 , and E 5 over F 2 . The groups E 3 (F 2 ) =
Z/5Z ={0,(0, 0), (1, 0), (0, 1), (1, 1)}, E 4 (F 2 ) = 0 ={0}, and E 5 (F 2 ) = Z/3Z
={0,(0, 0), (0, 1)}. Observe that the five elliptic curves could not be isomorphic
over F 2 in any sense preserving their structure as algebraic curves because all five
have different numbers of points over the field F 2 of two elements. In summary we
have:
(6.4) Proposition. Upto isomorphism over F 2 there are five elliptic curves defined
over F 2 . Two, namely E 1 and E 2 above, have j = 1 and three, namely E 3 , E 4 and
E 5 above, have j = 0.
Exercises
2
1. Show that the field of four elements F 4 will be of the form 0,1, u, u where u = u , uu =
2

1, 1 + u + u = 0, and 1 + u + u 2 = 1. “Graph” the curves E 1 ,... , E 5 of (6.4) over
the field of four elements. Show that E 1 and E 2 are isomorphic and E 3 and E 4 are
isomorphic over F 4 . Show that E 3 and E 5 are not isomorphic over F 4 .
2. Show that there is a field F 16 of 16 elements which is additively of the form F 16 =
2
F 4 + F 4 v where v + v = u. Show that every nonzero element is a square. Show that
there is a field F 256 of 256 elements which is additively of the form F 256 = F 16 + F 16 w
2 3
where w + w = v . Observe that we have the inclusions F 2 ⊂ F 4 ⊂ F 16 ⊂ F 256 and
F 2 is a two-dimensional vector space over F q .
q
3. Show that E 3 and E 5 are not isomorphic over F 16 , but that they are isomorphic over
F 256 .
4. Determine the automorphism groups Aut k (E i )with i = 1,... , 5 as in (6.3) and k = F q
for q = 2, 4, 16, 256.
5. Find all elliptic curves over F 4 up to isomorphism over F 4 . Which curves become iso-
morphic over F 16 ? Find their j-invariants.
6. Find all elliptic curves over F 3 up to isomorphism over F 3 . Show there are four with
j = 1or −1 and four with j = 0. Determine their groups of points over F 9 and which
ones are isomorphic over F 9 .
7. For an elliptic curve E over F 3 determine Aut k (E), where k = F 3 and F 9 .