| Curve name |
$X_{219b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{219}$ |
| Curves that $X_{219b}$ minimally covers |
|
| Curves that minimally cover $X_{219b}$ |
|
| Curves that minimally cover $X_{219b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -187596t^{24} + 18144t^{23} - 397008t^{22} - 2850336t^{21} -
565272t^{20} - 7299936t^{19} - 17609616t^{18} - 1956960t^{17} - 78137460t^{16} +
26847936t^{15} - 181169568t^{14} + 24373440t^{13} - 212260176t^{12} -
24373440t^{11} - 181169568t^{10} - 26847936t^{9} - 78137460t^{8} + 1956960t^{7}
- 17609616t^{6} + 7299936t^{5} - 565272t^{4} + 2850336t^{3} - 397008t^{2} -
18144t - 187596\]
\[B(t) = -31271184t^{36} + 4432320t^{35} - 97565472t^{34} - 727565760t^{33} -
13552272t^{32} - 3365079552t^{31} - 3983254272t^{30} - 12763381248t^{29} -
13853954880t^{28} - 50790212352t^{27} - 72542646144t^{26} - 54415348992t^{25} -
384962849856t^{24} + 99036504576t^{23} - 1011424548096t^{22} +
228023907840t^{21} - 1650526686432t^{20} + 144222996096t^{19} -
1922680601280t^{18} - 144222996096t^{17} - 1650526686432t^{16} -
228023907840t^{15} - 1011424548096t^{14} - 99036504576t^{13} -
384962849856t^{12} + 54415348992t^{11} - 72542646144t^{10} + 50790212352t^{9} -
13853954880t^{8} + 12763381248t^{7} - 3983254272t^{6} + 3365079552t^{5} -
13552272t^{4} + 727565760t^{3} - 97565472t^{2} - 4432320t - 31271184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 2316x - 42896$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |