| Curve name |
$X_{79c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 4 & 9 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 12 & 11 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{79c}$ minimally covers |
|
| Curves that minimally cover $X_{79c}$ |
|
| Curves that minimally cover $X_{79c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{12} - 7452t^{11} - 28944t^{10} - 65880t^{9} - 92340t^{8} -
80352t^{7} - 69120t^{6} - 160704t^{5} - 369360t^{4} - 527040t^{3} - 463104t^{2}
- 238464t - 57024\]
\[B(t) = 10206t^{18} + 127332t^{17} + 749412t^{16} + 2666736t^{15} +
5957712t^{14} + 6676992t^{13} - 6537888t^{12} - 49180608t^{11} - 125846784t^{10}
- 211545216t^{9} - 251693568t^{8} - 196722432t^{7} - 52303104t^{6} +
106831872t^{5} + 190646784t^{4} + 170671104t^{3} + 95924736t^{2} + 32596992t +
5225472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 26675x - 378250$, with conductor $5600$ |
| Generic density of odd order reductions |
$9249/57344$ |