| Curve name |
$X_{194j}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{194}$ |
| Curves that $X_{194j}$ minimally covers |
|
| Curves that minimally cover $X_{194j}$ |
|
| Curves that minimally cover $X_{194j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{24} - 9059696640t^{22} - 16533946368t^{20} -
12796821504t^{18} - 4862509056t^{16} - 4600627200t^{14} - 3791978496t^{12} -
287539200t^{10} - 18994176t^{8} - 3124224t^{6} - 252288t^{4} - 8640t^{2} - 108\]
\[B(t) = -29686813949952t^{36} - 222651104624640t^{34} - 684652146720768t^{32} -
1098412116148224t^{30} - 897562265518080t^{28} - 82798379532288t^{26} +
629714146295808t^{24} + 639078248742912t^{22} + 231497898393600t^{20} +
25209058885632t^{18} + 14468618649600t^{16} + 2496399409152t^{14} +
153738805248t^{12} - 1263403008t^{10} - 855982080t^{8} - 65470464t^{6} -
2550528t^{4} - 51840t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 2086586x - 1086607584$, with conductor $8670$ |
| Generic density of odd order reductions |
$271/2688$ |