| Curve name |
$X_{207c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207c}$ minimally covers |
|
| Curves that minimally cover $X_{207c}$ |
|
| Curves that minimally cover $X_{207c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} - 4752t^{21} + 8208t^{20} + 2989440t^{19} + 62339328t^{18} +
638337024t^{17} + 3673755648t^{16} + 10080681984t^{15} - 12150079488t^{14} -
221109682176t^{13} - 894669815808t^{12} - 1768877457408t^{11} -
777605087232t^{10} + 5161309175808t^{9} + 15047703134208t^{8} +
20917027602432t^{7} + 16341880799232t^{6} + 6269310074880t^{5} +
137707388928t^{4} - 637802643456t^{3} - 115964116992t^{2}\]
\[B(t) = 432t^{33} + 28512t^{32} + 1757376t^{31} + 71245440t^{30} +
1694836224t^{29} + 24404944896t^{28} + 205952827392t^{27} + 662685401088t^{26} -
6586979844096t^{25} - 110131636469760t^{24} - 813424020553728t^{23} -
3662208293142528t^{22} - 9354237853040640t^{21} - 1998858309599232t^{20} +
89587256867684352t^{19} + 381431083394138112t^{18} + 716698054941474816t^{17} -
127926931814350848t^{16} - 4789369780756807680t^{15} -
15000405168711794688t^{14} - 26654278305504559104t^{13} -
28870347710728765440t^{12} - 13813897954005614592t^{11} +
11118016114100011008t^{10} + 27642520567730405376t^{9} +
26204610047250530304t^{8} + 14558532308312260608t^{7} + 4895949356626083840t^{6}
+ 966127673187237888t^{5} + 125397102124597248t^{4} + 15199648742375424t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 731508x + 10909262576$, with conductor $20160$ |
| Generic density of odd order reductions |
$11/112$ |