| Curve name |
$X_{235r}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 15 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{235}$ |
| Curves that $X_{235r}$ minimally covers |
|
| Curves that minimally cover $X_{235r}$ |
|
| Curves that minimally cover $X_{235r}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{28} + 432t^{27} + 648t^{26} - 3888t^{25} + 540t^{24} +
7776t^{23} - 3888t^{22} + 6048t^{21} - 2700t^{20} - 24624t^{19} + 6264t^{18} +
7344t^{17} + 2268t^{16} + 25920t^{15} - 6048t^{14} - 25920t^{13} + 2268t^{12} -
7344t^{11} + 6264t^{10} + 24624t^{9} - 2700t^{8} - 6048t^{7} - 3888t^{6} -
7776t^{5} + 540t^{4} + 3888t^{3} + 648t^{2} - 432t - 108\]
\[B(t) = 432t^{42} - 2592t^{41} - 1296t^{40} + 32832t^{39} - 33696t^{38} -
124416t^{37} + 201312t^{36} + 77760t^{35} - 193104t^{34} + 379296t^{33} -
695952t^{32} - 642816t^{31} + 1302912t^{30} - 165888t^{29} + 777600t^{28} +
808704t^{27} - 2304288t^{26} - 295488t^{25} - 676512t^{24} - 51840t^{23} +
1622592t^{22} + 1622592t^{20} + 51840t^{19} - 676512t^{18} + 295488t^{17} -
2304288t^{16} - 808704t^{15} + 777600t^{14} + 165888t^{13} + 1302912t^{12} +
642816t^{11} - 695952t^{10} - 379296t^{9} - 193104t^{8} - 77760t^{7} +
201312t^{6} + 124416t^{5} - 33696t^{4} - 32832t^{3} - 1296t^{2} + 2592t + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 376750x + 88826500$, with conductor $1050$ |
| Generic density of odd order reductions |
$1091/10752$ |