| Curve name |
$X_{79b}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{79b}$ minimally covers |
|
| Curves that minimally cover $X_{79b}$ |
|
| Curves that minimally cover $X_{79b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3564t^{8} - 15552t^{7} - 25056t^{6} - 10368t^{5} + 11232t^{4} -
20736t^{3} - 100224t^{2} - 124416t - 57024\]
\[B(t) = 81648t^{12} + 528768t^{11} + 1353024t^{10} + 1085184t^{9} -
3063744t^{8} - 12026880t^{7} - 21772800t^{6} - 24053760t^{5} - 12254976t^{4} +
8681472t^{3} + 21648384t^{2} + 16920576t + 5225472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 52268x + 4599408$, with conductor $2240$ |
| Generic density of odd order reductions |
$419/2688$ |