| Curve name |
$X_{101i}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{101}$ |
| Curves that $X_{101i}$ minimally covers |
|
| Curves that minimally cover $X_{101i}$ |
|
| Curves that minimally cover $X_{101i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -28311552t^{12} + 21233664t^{10} - 6193152t^{8} + 884736t^{6} -
69120t^{4} + 3456t^{2} - 108\]
\[B(t) = 57982058496t^{18} - 65229815808t^{16} + 31255953408t^{14} -
8323596288t^{12} + 1316487168t^{10} - 116785152t^{8} + 3870720t^{6} +
207360t^{4} - 20736t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 3431x + 59961$, with conductor $1470$ |
| Generic density of odd order reductions |
$193/1792$ |