| Curve name |
$X_{37a}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 14 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{37}$ |
| Curves that $X_{37a}$ minimally covers |
|
| Curves that minimally cover $X_{37a}$ |
|
| Curves that minimally cover $X_{37a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -49545216t^{12} - 191102976t^{11} - 332660736t^{10} - 353009664t^{9} -
258453504t^{8} - 138682368t^{7} - 56180736t^{6} - 17335296t^{5} - 4038336t^{4} -
689472t^{3} - 81216t^{2} - 5832t - 189\]
\[B(t) = 123211874304t^{18} + 684913065984t^{17} + 1747615481856t^{16} +
2723344809984t^{15} + 2897970462720t^{14} + 2216794521600t^{13} +
1236960018432t^{12} + 488968814592t^{11} + 117252292608t^{10} - 14656536576t^{8}
- 7640137728t^{7} - 2415937536t^{6} - 541209600t^{5} - 88439040t^{4} -
10388736t^{3} - 833328t^{2} - 40824t - 918\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1658x + 25438$, with conductor $3200$ |
| Generic density of odd order reductions |
$85091/344064$ |