| Curve name |
$X_{57a}$ |
| Index |
$32$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 7 \\ 1 & 2 \end{matrix}\right],
\left[ \begin{matrix} 15 & 11 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{57}$ |
| Curves that $X_{57a}$ minimally covers |
|
| Curves that minimally cover $X_{57a}$ |
|
| Curves that minimally cover $X_{57a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -32925150t^{20} + 16167384t^{19} + 101160468t^{18} + 230758632t^{17} +
336911994t^{16} + 315360t^{15} - 955654416t^{14} - 2522286432t^{13} -
4398721308t^{12} - 5861269296t^{11} - 6326184456t^{10} - 5664670416t^{9} -
4206560796t^{8} - 2560107168t^{7} - 1246787856t^{6} - 468036576t^{5} -
129264390t^{4} - 24870888t^{3} - 3100140t^{2} - 221400t - 6750\]
\[B(t) = -72719803296t^{30} + 53607412224t^{29} + 328107793824t^{28} +
684074101248t^{27} + 652759044192t^{26} - 1519490534400t^{25} -
6722743710816t^{24} - 13362326590464t^{23} - 15937371346464t^{22} -
3748273740288t^{21} + 34341605284896t^{20} + 102487396953600t^{19} +
193133174982624t^{18} + 285794906038272t^{17} + 353849310717984t^{16} +
376978275846144t^{15} + 350002399786272t^{14} + 284385380212224t^{13} +
202153174921440t^{12} + 125191778674176t^{11} + 66960218981664t^{10} +
30553319605248t^{9} + 11732785809120t^{8} + 3741616032768t^{7} +
977369185440t^{6} + 205582551552t^{5} + 33963564384t^{4} + 4237263360t^{3} +
374090400t^{2} + 20736000t + 540000\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 333x - 6037$, with conductor $6400$ |
| Generic density of odd order reductions |
$977931/1835008$ |