| Curve name |
$X_{192m}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192m}$ minimally covers |
|
| Curves that minimally cover $X_{192m}$ |
|
| Curves that minimally cover $X_{192m}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 95040t^{22} + 3480192t^{20} + 8100864t^{18} -
1107219456t^{16} + 7029227520t^{14} - 25237979136t^{12} + 112467640320t^{10} -
283448180736t^{8} + 33181138944t^{6} + 228077862912t^{4} - 99656663040t^{2} -
1811939328\]
\[B(t) = 432t^{36} - 922752t^{34} - 14867712t^{32} + 4087037952t^{30} -
121712689152t^{28} + 1124205723648t^{26} - 2462644961280t^{24} +
59765120040960t^{22} - 881211834630144t^{20} + 5063543724441600t^{18} -
14099389354082304t^{16} + 15299870730485760t^{14} - 10086993761402880t^{12} +
73675946304995328t^{10} - 127625004740247552t^{8} + 68569118520901632t^{6} -
3991021050396672t^{4} - 3963189662318592t^{2} + 29686813949952\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 376476865x + 2811485935775$, with conductor $47040$ |
| Generic density of odd order reductions |
$271/2688$ |