| Curve name |
$X_{207l}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207l}$ minimally covers |
|
| Curves that minimally cover $X_{207l}$ |
|
| Curves that minimally cover $X_{207l}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 864t^{15} + 13824t^{14} + 628992t^{13} + 7402752t^{12} +
36771840t^{11} + 30965760t^{10} - 514473984t^{9} - 2477924352t^{8} -
4115791872t^{7} + 1981808640t^{6} + 18827182080t^{5} + 30321672192t^{4} +
20610809856t^{3} + 3623878656t^{2} - 1811939328t - 452984832\]
\[B(t) = -54t^{24} - 2592t^{23} - 165888t^{22} - 5550336t^{21} - 88687872t^{20}
- 635185152t^{19} + 371589120t^{18} + 45072433152t^{17} + 383285551104t^{16} +
1506238267392t^{15} + 1640258076672t^{14} - 11447323066368t^{13} -
57189844647936t^{12} - 91578584530944t^{11} + 104976516907008t^{10} +
771193992904704t^{9} + 1569937617321984t^{8} + 1476933489524736t^{7} +
97409858273280t^{6} - 1332079811887104t^{5} - 1487935585124352t^{4} -
744953487556608t^{3} - 178120883699712t^{2} - 22265110462464t - 3710851743744\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 20320x + 50499072$, with conductor $1680$ |
| Generic density of odd order reductions |
$19/336$ |