| Curve name |
$X_{57b}$ |
| Index |
$32$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 31 & 27 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 17 & 23 \\ 1 & 2 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{57}$ |
| Curves that $X_{57b}$ minimally covers |
|
| Curves that minimally cover $X_{57b}$ |
|
| Curves that minimally cover $X_{57b}$ 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 |
$977931/1835008$ |