| Curve name |
$X_{213b}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 24 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213b}$ minimally covers |
|
| Curves that minimally cover $X_{213b}$ |
|
| Curves that minimally cover $X_{213b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{30} - 2376t^{28} - 20196t^{26} - 82512t^{24} - 161676t^{22} -
115128t^{20} + 71388t^{18} + 178848t^{16} + 71388t^{14} - 115128t^{12} -
161676t^{10} - 82512t^{8} - 20196t^{6} - 2376t^{4} - 108t^{2}\]
\[B(t) = 432t^{45} + 14256t^{43} + 199584t^{41} + 1540512t^{39} + 7121520t^{37}
+ 19936368t^{35} + 31653504t^{33} + 19460736t^{31} - 21319200t^{29} -
54966816t^{27} - 60264000t^{25} - 60264000t^{23} - 54966816t^{21} -
21319200t^{19} + 19460736t^{17} + 31653504t^{15} + 19936368t^{13} +
7121520t^{11} + 1540512t^{9} + 199584t^{7} + 14256t^{5} + 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 205503125x + 1132671382125$, with conductor
$252150$ |
| Generic density of odd order reductions |
$51/448$ |