Author: William A. Stein
Curve Examples

# Curve Examples

## Rank 5 Example

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:

[[0, 0, 1, -79, 342], -19047851, odd, 5, 2.047641, 30.285711, 14.790539, 5, 14.790528, [ 5, 4, 3, 7, 0] ],

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:

[[0, 0, 1, -79, 342], -19047851, odd, 5, 2.047641, 30.285711, 14.790539, 5, 14.790528, [ -10, -8, -7, -6, -3, -1, 0, 3, 4, 5, 7, 10, 12, 14, 19, 33, 38, 43, 45, 62, 88, 92, 97, 199, 265, 315, 434, 488, 543, 1495, 1522, 2040, 2317, 5542, 10120, 38624, 133767, 180445 ] ],
but we chose just to keep a basis of the points found.

## Setzer-Neumann Curve with Tate-Shafarevich group of order 289

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

[[1, 1, 0, -103324, -12826653], 4959593, setzer, 0, 0.266286, 19.239175, 288.999919, 0, 1.000000, [] ],

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.


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