| Curve name |
$X_{101f}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{101}$ |
| Curves that $X_{101f}$ minimally covers |
|
| Curves that minimally cover $X_{101f}$ |
|
| Curves that minimally cover $X_{101f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -113246208t^{16} + 141557760t^{14} - 74317824t^{12} + 21233664t^{10} -
3594240t^{8} + 373248t^{6} - 24624t^{4} + 1080t^{2} - 27\]
\[B(t) = 463856467968t^{24} - 869730877440t^{22} + 728399609856t^{20} -
359216971776t^{18} + 115511132160t^{16} - 25225592832t^{14} + 3746856960t^{12} -
361304064t^{10} + 18994176t^{8} - 44928t^{6} - 59616t^{4} + 3240t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 345891x + 78186850$, with conductor $7056$ |
| Generic density of odd order reductions |
$41/336$ |