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§4. Elliptic Curves over the Real Numbers 289

Table 2.
Table of Elliptic Curves. Sixteen Curves in Serre’s Inventiones article [1972], pp.
309, 310, 315–316, 318–319.
Semistable case
2 12
2 3 2
5.5.1 y + y = x − x N = 11 =−11 j =−
11
∼ 1 (11)
−2 12 · 31 3
2 5
5.5.2 y + y n = 11 =−11 j =
11 5
2
3
∼ 0 (11) = x − x − 10x − 20
2 12 · 3 3
2
3
5.5.6 y + y = x − x N = 37 = 37 j =
37
−2 12
3
2
5.5.7 y + y = x + x 2 N = 43 =−43 j =
43
3
−3 · 5 3
2
3
5.5.8 y + xy + y = x − x 2 N = 53 = 53 j =
53
−5 6
2
3
2
5.5.3 y + xy + y = x − x N = 2 · 7 =−2 · 7 j =
2
2 · 7
3
−3 · 43 3
7
2
5.5.4 y + xy + y N = 2 · 13 =−2 · 13 j =
7
2 · 13
3 2
= x − x − 3x + 3
−193 3
2 8 2
5.5.5 y + xy + y N = 2 · 3 · 7 =−2 · 3 · 7 j =
2
8
2 · 3 · 7
3
2
= x + x − 4x + 5
Not semistable; j nonintegral
2 14
2 3 2 2 4
5.7.1 y = x + x − x N = 2 · 5 = 2 · 5 j =
5
2 11
3
4
3
2
2
5.7.2 y = x − x + x N = 2 · 3 =−2 · 3 j =
3
−1
2 3 2 2 7
5.7.3 y + xy = x − x − 5 N = 3 · 3 =−3 · 5 j =
3 · 5
−5 · 29 3
5
2
5.7.4 y + xy + y N = 2 · 5 2 =−2 · 5 2 j =
2 5
3
2
= x + x − 3x + 1
Not semistable; j integral
5
3
2
2
5.9.1 y = x − 2x − x N = 2 7 = 2 7 j = 2 · 7 3
2
6
9
6
3
5.9.2 y = x + 6x − 2 N = 2 · 3 3 =−2 · 3 5 j = 2 · 3
3
2
5.9.3 y + xy N = 7 2 =−7 3 j =−3 · 5 3
3
2
= x − x − 2x − 1
2
5.9.4 y + xy N = 11 2 =−11 4 j =−11 2
2
3
= x − x − 2x − 7