Elliptic curves in class 4.1-a over 4.4.13068.1
Isogeny class 4.1-a contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 4.1-a1
| \( \bigl[a^{2} - 2 a - 3\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 0\) , \( -38 a^{3} + 58 a^{2} + 194 a - 69\) , \( 108 a^{3} - 169 a^{2} - 559 a + 198\bigr] \)
|
| 4.1-a2
| \( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( -7 a^{3} - 14 a^{2} - a + 1\) , \( -7 a^{3} - 10 a^{2} + 5 a - 1\bigr] \)
|
| 4.1-a3
| \( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( -72 a^{3} - 159 a^{2} - 41 a + 16\) , \( 1682 a^{3} + 3425 a^{2} + 338 a - 572\bigr] \)
|
| 4.1-a4
| \( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( -2 a^{3} + 3 a^{2} + 9 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( -4 a^{3} + 6 a^{2} + 18 a - 4\) , \( 14 a^{3} - 22 a^{2} - 72 a + 26\bigr] \)
|
| 4.1-a5
| \( \bigl[a^{3} - a^{2} - 6 a\) , \( -2 a^{3} + 3 a^{2} + 9 a - 1\) , \( a + 1\) , \( 147 a^{3} - 102 a^{2} - 929 a - 453\) , \( 1766 a^{3} - 1166 a^{2} - 10943 a - 5358\bigr] \)
|
| 4.1-a6
| \( \bigl[a^{3} - a^{2} - 6 a\) , \( -2 a^{3} + 3 a^{2} + 9 a - 1\) , \( a + 1\) , \( 12 a^{3} - 7 a^{2} - 79 a - 38\) , \( 15 a^{3} - 9 a^{2} - 95 a - 47\bigr] \)
|
| 4.1-a7
| \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -34 a^{3} + 52 a^{2} + 174 a - 61\) , \( 16 a^{3} - 25 a^{2} - 83 a + 29\bigr] \)
|
| 4.1-a8
| \( \bigl[a + 1\) , \( 2 a^{3} - 3 a^{2} - 10 a + 1\) , \( a + 1\) , \( 2 a^{3} - 3 a^{2} - 10 a + 4\) , \( a^{3} - a^{2} - 5 a + 1\bigr] \)
|
Rank: \( 1 \)
\(\left(\begin{array}{rrrrrrrr}
1 & 2 & 4 & 3 & 12 & 6 & 12 & 4 \\
2 & 1 & 2 & 6 & 6 & 3 & 6 & 2 \\
4 & 2 & 1 & 12 & 3 & 6 & 12 & 4 \\
3 & 6 & 12 & 1 & 4 & 2 & 4 & 12 \\
12 & 6 & 3 & 4 & 1 & 2 & 4 & 12 \\
6 & 3 & 6 & 2 & 2 & 1 & 2 & 6 \\
12 & 6 & 12 & 4 & 4 & 2 & 1 & 3 \\
4 & 2 & 4 & 12 & 12 & 6 & 3 & 1
\end{array}\right)\)
