| Curve name |
$X_{79f}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{79f}$ minimally covers |
|
| Curves that minimally cover $X_{79f}$ |
|
| Curves that minimally cover $X_{79f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{16} - 11016t^{15} - 51624t^{14} - 100656t^{13} + 8208t^{12} +
384480t^{11} + 538272t^{10} - 278208t^{9} - 1364256t^{8} - 556416t^{7} +
2153088t^{6} + 3075840t^{5} + 131328t^{4} - 3220992t^{3} - 3303936t^{2} -
1410048t - 228096\]
\[B(t) = 10206t^{24} + 188568t^{23} + 1452168t^{22} + 5746032t^{21} +
9924768t^{20} - 11205216t^{19} - 99946656t^{18} - 222201792t^{17} -
143589024t^{16} + 433057536t^{15} + 1337990400t^{14} + 1579378176t^{13} -
3158756352t^{11} - 5351961600t^{10} - 3464460288t^{9} + 2297424384t^{8} +
7110457344t^{7} + 6396585984t^{6} + 1434267648t^{5} - 2540740608t^{4} -
2941968384t^{3} - 1487020032t^{2} - 386187264t - 41803776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 640283x - 197199618$, with conductor $7840$ |
| Generic density of odd order reductions |
$149/896$ |