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Here is the first rank 5 elliptic curve that we found, from 001_3, note there is also a rank 2 curve of the same conductor in the same file:

One program we wrote interprets the format in a "user-friendly" way, here is what we get as a report, with a little extra computation thrown in:

The elliptic curve [0, 0, 1, -79, 342] has equation: $y^2 + y = x^3 -79 x +342$. The invariants are: b2 = 0, b4 = -158, b6 = 1369, b8 = -6241, c4 = 3792, c6 = -295704. The discriminant is -19047851. The predicted rank parity is odd. The number of components is 1. The real two-division point is -10.555718. The real period is 2.047641. The analytic rank is: 5. The L-series value (or derivative) is: 30.285711. Then the predicted regulator is: 14.790539. The point rank is: 5. The regulator is: 14.790528. The analytic and point ranks agree. The quotient of the L-value by the period and regulator is: 1.000001. Thus the predicted size of the Tate-Shafarevich group is: 1. The independent points have x-coordinates: 5, 4, 3, 7, 0. Here are the points with their heights. P_1 = [5, 8, 1] has height 1.052241. P_1 + P_2 = [-8, -22, 1] has height 1.342677. P_1 - P_2 = [315, -5589, 1] has height 2.880110. P_1 + P_3 = [-46, -201, 8] has height 1.993585. P_1 - P_3 = [92, -879, 1] has height 2.273482. P_1 + P_4 = [-78, 105, 8] has height 2.065716. P_1 - P_4 = [88, 821, 1] has height 2.251796. P_1 + P_5 = [-1, -21, 1] has height 1.192578. P_1 - P_5 = [3020, -14058, 125] has height 3.243683. P_2 = [4, 9, 1] has height 1.059152. P_2 + P_3 = [-3, -24, 1] has height 1.241578. P_2 - P_3 = [434, -9040, 1] has height 3.039310. P_2 + P_4 = [-285, -8, 27] has height 2.484190. P_2 - P_4 = [38, 228, 1] has height 1.847143. P_2 + P_5 = [68, -1063, 64] has height 2.522368. P_2 - P_5 = [45, -297, 1] has height 1.927714. P_3 = [3, 11, 1] has height 1.081292. P_3 + P_4 = [-10, -12, 1] has height 1.376653. P_3 - P_4 = [1476, 6615, 64] has height 2.998960. P_3 + P_5 = [66, -359, 27] has height 2.195034. P_3 - P_5 = [97, -952, 1] has height 2.299328. P_4 = [7, 11, 1] has height 1.106514. P_4 + P_5 = [-6, -25, 1] has height 1.305368. P_4 - P_5 = [3899, -10536, 343] has height 3.239439. P_5 = [0, 18, 1] has height 1.165889.

Of course, this particular curve has many integral points:

As another example of the notation used, here is the example (from 110_1):

The verbose display for this curve is:

The elliptic curve [1, 1, 0, -103324, -12826653] has equation: $y^2 + xy = x^3 + x^2 -103324 x -12826653$ The invariants are: b2 = 5, b4 = -206648, b6 = -51306612, b8 = -10739982241, c4 = 4959577, c6 = 11045031427. Discriminant is 4959593 Rank parity is setzer Number of components is 2 The two-division points are -186.001796, -186.000000, 370.751796 Real period is 0.266286 The analytic rank is: 0 The L-series value (or derivative) is: 19.239175 Dividing this by the period gives the predicted regulator: 288.999919 The point rank is: 0 The regulator is: 1.000000 The analytic and point ranks agree. The quotient of the L-value by the period and regulator is: 288.999919 Thus the predicted size of the Tate-Shafarevich group is: 289 No points were found.

Page created on a Macintosh Performa 6205CD, using MPW, MacPerl, and OzTeX, Sun Apr 18 23:51:22 1999.

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