| Curve name |
$X_{102d}$ |
| 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 \\ 0 & 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 $X_{102d}$ minimally covers |
|
| Curves that minimally cover $X_{102d}$ |
|
| Curves that minimally cover $X_{102d}$ 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 |
$y^2 + xy = x^3 - x^2 - 17208x + 867901$, with conductor $441$ |
| Generic density of odd order reductions |
$25/224$ |