| Curve name |
$X_{94e}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{94}$ |
| Curves that $X_{94e}$ minimally covers |
|
| Curves that minimally cover $X_{94e}$ |
|
| Curves that minimally cover $X_{94e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 324t^{16} + 10368t^{15} + 117504t^{14} + 387072t^{13} - 2875392t^{12} -
28753920t^{11} - 65912832t^{10} + 199950336t^{9} + 1299677184t^{8} +
1599602688t^{7} - 4218421248t^{6} - 14722007040t^{5} - 11777605632t^{4} +
12683575296t^{3} + 30802968576t^{2} + 21743271936t + 5435817984\]
\[B(t) = 31104t^{23} + 1430784t^{22} + 27841536t^{21} + 287981568t^{20} +
1530814464t^{19} + 1571733504t^{18} - 30125260800t^{17} - 170378919936t^{16} -
155062370304t^{15} + 1805201178624t^{14} + 6837579546624t^{13} -
54700636372992t^{11} - 115532875431936t^{10} + 79391933595648t^{9} +
697872056057856t^{8} + 987144545894400t^{7} - 412020507672576t^{6} -
3210350614806528t^{5} - 4831528970354688t^{4} - 3736827705950208t^{3} -
1536292621910016t^{2} - 267181325549568t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 28028x - 4961152$, with conductor $7840$ |
| Generic density of odd order reductions |
$149/896$ |