| Curve name |
$X_{76c}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{76}$ |
| Curves that $X_{76c}$ minimally covers |
|
| Curves that minimally cover $X_{76c}$ |
|
| Curves that minimally cover $X_{76c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -2430t^{20} + 47952t^{19} + 227664t^{18} - 482112t^{17} + 3227040t^{16}
+ 13353984t^{15} + 3898368t^{14} + 67350528t^{13} - 210263040t^{12} +
1371561984t^{11} - 1515331584t^{10} - 5486247936t^{9} - 3364208640t^{8} -
4310433792t^{7} + 997982208t^{6} - 13674479616t^{5} + 13217955840t^{4} +
7898923008t^{3} + 14920187904t^{2} - 12570329088t - 2548039680\]
\[B(t) = 198288t^{30} + 933120t^{29} + 3810240t^{28} + 5197824t^{27} -
178972416t^{26} + 731566080t^{25} + 952777728t^{24} - 33570422784t^{23} +
32805015552t^{22} + 60436316160t^{21} - 1389027999744t^{20} + 91814363136t^{19}
- 1315314008064t^{18} - 11691680071680t^{17} - 27920845504512t^{16} -
111683382018048t^{14} + 187066881146880t^{13} - 84180096516096t^{12} -
23504476962816t^{11} - 1422364671737856t^{10} - 247547150991360t^{9} +
537477374803968t^{8} + 2200071227572224t^{7} + 249764964728832t^{6} -
767102633902080t^{5} - 750664720318464t^{4} - 87205015977984t^{3} +
255700877967360t^{2} - 250482492702720t + 212910118797312\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 120x - 2176$, with conductor $2304$ |
| Generic density of odd order reductions |
$403/1792$ |