| Curve name |
$X_{195i}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{195}$ |
| Curves that $X_{195i}$ minimally covers |
|
| Curves that minimally cover $X_{195i}$ |
|
| Curves that minimally cover $X_{195i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{24} + 1714176t^{20} - 1762560t^{16} + 670464t^{12} -
110160t^{8} + 6696t^{4} - 27\]
\[B(t) = 14155776t^{36} + 435290112t^{32} - 1252786176t^{28} + 1164312576t^{24}
- 529846272t^{20} + 132461568t^{16} - 18192384t^{12} + 1223424t^{8} - 26568t^{4}
- 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 315x + 1066$, with conductor $45$ |
| Generic density of odd order reductions |
$299/2688$ |