| Curve name |
$X_{214b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{214}$ |
| Curves that $X_{214b}$ minimally covers |
|
| Curves that minimally cover $X_{214b}$ |
|
| Curves that minimally cover $X_{214b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -81t^{16} - 1296t^{15} - 12096t^{14} - 78624t^{13} - 341712t^{12} -
955584t^{11} - 1693440t^{10} - 1994112t^{9} - 2261088t^{8} - 3988224t^{7} -
6773760t^{6} - 7644672t^{5} - 5467392t^{4} - 2515968t^{3} - 774144t^{2} -
165888t - 20736\]
\[B(t) = 3888t^{23} + 89424t^{22} + 959904t^{21} + 6386688t^{20} +
29937600t^{19} + 108706752t^{18} + 330625152t^{17} + 868589568t^{16} +
1891524096t^{15} + 3090700800t^{14} + 3078964224t^{13} - 6157928448t^{11} -
12362803200t^{10} - 15132192768t^{9} - 13897433088t^{8} - 10580004864t^{7} -
6957232128t^{6} - 3832012800t^{5} - 1634992128t^{4} - 491470848t^{3} -
91570176t^{2} - 7962624t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 428257x + 84690144$, with conductor $84320$ |
| Generic density of odd order reductions |
$9827/86016$ |