| Curve name |
$X_{102c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102c}$ minimally covers |
|
| Curves that minimally cover $X_{102c}$ |
|
| Curves that minimally cover $X_{102c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{14} - 162t^{13} + 621t^{12} + 2808t^{11} - 10584t^{10} -
7776t^{9} + 79920t^{8} - 129600t^{7} + 21168t^{6} + 227232t^{5} - 404784t^{4} +
362880t^{3} - 190080t^{2} + 55296t - 6912\]
\[B(t) = -54t^{21} - 486t^{20} + 1134t^{19} + 14742t^{18} - 27216t^{17} -
191160t^{16} + 565920t^{15} + 751680t^{14} - 5338224t^{13} + 6697296t^{12} +
8746704t^{11} - 43311024t^{10} + 82418688t^{9} - 117319104t^{8} + 150450048t^{7}
- 169845120t^{6} + 152368128t^{5} - 100818432t^{4} + 46835712t^{3} -
14432256t^{2} + 2654208t - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 275331x - 55270334$, with conductor $7056$ |
| Generic density of odd order reductions |
$41/336$ |