| Curve name |
$X_{79j}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \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 $X_{79j}$ minimally covers |
|
| Curves that minimally cover $X_{79j}$ |
|
| Curves that minimally cover $X_{79j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3564t^{16} - 44064t^{15} - 206496t^{14} - 402624t^{13} + 32832t^{12} +
1537920t^{11} + 2153088t^{10} - 1112832t^{9} - 5457024t^{8} - 2225664t^{7} +
8612352t^{6} + 12303360t^{5} + 525312t^{4} - 12883968t^{3} - 13215744t^{2} -
5640192t - 912384\]
\[B(t) = 81648t^{24} + 1508544t^{23} + 11617344t^{22} + 45968256t^{21} +
79398144t^{20} - 89641728t^{19} - 799573248t^{18} - 1777614336t^{17} -
1148712192t^{16} + 3464460288t^{15} + 10703923200t^{14} + 12635025408t^{13} -
25270050816t^{11} - 42815692800t^{10} - 27715682304t^{9} + 18379395072t^{8} +
56883658752t^{7} + 51172687872t^{6} + 11474141184t^{5} - 20325924864t^{4} -
23535747072t^{3} - 11896160256t^{2} - 3089498112t - 334430208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 209132x + 8303344$, with conductor $15680$ |
| Generic density of odd order reductions |
$419/2688$ |