| Curve name |
$X_{234e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{234}$ |
| Curves that $X_{234e}$ minimally covers |
|
| Curves that minimally cover $X_{234e}$ |
|
| Curves that minimally cover $X_{234e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1120716t^{24} - 20875968t^{23} - 197335872t^{22} - 1218454272t^{21} -
5360591232t^{20} - 17753637888t^{19} - 45664625664t^{18} - 94683893760t^{17} -
167975285760t^{16} - 249280561152t^{15} - 401251663872t^{14} -
327865466880t^{13} - 1214252384256t^{12} + 1311461867520t^{11} -
6420026621952t^{10} + 15953955913728t^{9} - 43001673154560t^{8} +
96956307210240t^{7} - 187042306719744t^{6} + 290875603156992t^{5} -
351311706980352t^{4} + 319410476679168t^{3} - 206921659318272t^{2} +
87560156086272t - 18802494406656\]
\[B(t) = 456657264t^{36} + 12759576192t^{35} + 180029609856t^{34} +
1683639461376t^{33} + 11578847346432t^{32} + 61906260344832t^{31} +
266105578143744t^{30} + 942417547296768t^{29} + 2803299591831552t^{28} +
7138223020965888t^{27} + 15844713734209536t^{26} + 31288075397627904t^{25} +
56005863392673792t^{24} + 92221034006052864t^{23} + 143524338497224704t^{22} +
199191383922180096t^{21} + 300132618347741184t^{20} + 230696874896523264t^{19} +
781818113883635712t^{18} - 922787499586093056t^{17} + 4802121893563858944t^{16}
- 12748248571019526144t^{15} + 36742230655289524224t^{14} -
94434338822198132736t^{13} + 229400016456391852032t^{12} -
512623827314735579136t^{11} + 1038399159285156151296t^{10} -
1871242335608081743872t^{9} + 2939472672804361469952t^{8} -
3952785688297023209472t^{7} + 4464510763322472136704t^{6} -
4154458806229923790848t^{5} + 3108173167393863892992t^{4} -
1807794106216243789824t^{3} + 773221286643159269376t^{2} -
219207849821840867328t + 31381248229773410304\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 13836x + 626416$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |