| Curve name |
$X_{192j}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192j}$ minimally covers |
|
| Curves that minimally cover $X_{192j}$ |
|
| Curves that minimally cover $X_{192j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} - 99360t^{20} - 464832t^{18} + 16754688t^{16} -
105781248t^{14} + 420691968t^{12} - 1692499968t^{10} + 4289200128t^{8} -
1903951872t^{6} - 6511656960t^{4} - 113246208t^{2}\]
\[B(t) = 432t^{33} - 896832t^{31} - 69113088t^{29} + 844729344t^{27} -
2322763776t^{25} + 25245499392t^{23} - 542808539136t^{21} + 4414542446592t^{19}
- 17658169786368t^{17} + 34739746504704t^{15} - 25851391377408t^{13} +
38056161705984t^{11} - 221440729153536t^{9} + 289881301450752t^{7} +
60185376718848t^{5} - 463856467968t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 69148812x + 221322182384$, with conductor $20160$ |
| Generic density of odd order reductions |
$11/112$ |