| Curve name |
$X_{94b}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{94}$ |
| Curves that $X_{94b}$ minimally covers |
|
| Curves that minimally cover $X_{94b}$ |
|
| Curves that minimally cover $X_{94b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 81t^{16} + 2592t^{15} + 29376t^{14} + 96768t^{13} - 718848t^{12} -
7188480t^{11} - 16478208t^{10} + 49987584t^{9} + 324919296t^{8} + 399900672t^{7}
- 1054605312t^{6} - 3680501760t^{5} - 2944401408t^{4} + 3170893824t^{3} +
7700742144t^{2} + 5435817984t + 1358954496\]
\[B(t) = 3888t^{23} + 178848t^{22} + 3480192t^{21} + 35997696t^{20} +
191351808t^{19} + 196466688t^{18} - 3765657600t^{17} - 21297364992t^{16} -
19382796288t^{15} + 225650147328t^{14} + 854697443328t^{13} -
6837579546624t^{11} - 14441609428992t^{10} + 9923991699456t^{9} +
87234007007232t^{8} + 123393068236800t^{7} - 51502563459072t^{6} -
401293826850816t^{5} - 603941121294336t^{4} - 467103463243776t^{3} -
192036577738752t^{2} - 33397665693696t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 7007x + 620144$, with conductor $15680$ |
| Generic density of odd order reductions |
$419/2688$ |