| Curve name |
$X_{227d}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 3 \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 |
$X_{227}$ |
| Curves that $X_{227d}$ minimally covers |
|
| Curves that minimally cover $X_{227d}$ |
|
| Curves that minimally cover $X_{227d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -115964116992t^{32} + 3478923509760t^{30} - 3913788948480t^{28} +
1522029035520t^{26} - 494659436544t^{24} + 67947724800t^{22} + 15288238080t^{20}
- 11041505280t^{18} + 3868065792t^{16} - 690094080t^{14} + 59719680t^{12} +
16588800t^{10} - 7547904t^{8} + 1451520t^{6} - 233280t^{4} + 12960t^{2} - 27\]
\[B(t) = 15199648742375424t^{48} + 957577870769651712t^{46} -
3950008716924813312t^{44} + 3650765632309297152t^{42} -
1977141809066803200t^{40} + 714442864519544832t^{38} - 169726937055363072t^{36}
+ 2571620258414592t^{34} + 15445724598632448t^{32} - 7568745987833856t^{30} +
2290349292650496t^{28} - 363460533682176t^{26} + 22716283355136t^{22} -
8946676924416t^{20} + 1847838375936t^{18} - 235683053568t^{16} -
2452488192t^{14} + 10116513792t^{12} - 2661507072t^{10} + 460339200t^{8} -
53125632t^{6} + 3592512t^{4} - 54432t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 + 425912395x + 15999991513647$, with conductor
$130050$ |
| Generic density of odd order reductions |
$299/2688$ |