| Curve name |
$X_{197b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{197}$ |
| Curves that $X_{197b}$ minimally covers |
|
| Curves that minimally cover $X_{197b}$ |
|
| Curves that minimally cover $X_{197b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 45684t^{24} - 1591488t^{23} - 18591552t^{22} - 72997632t^{21} -
227187072t^{20} - 690398208t^{19} - 3708232704t^{18} - 12310548480t^{17} -
18470384640t^{16} - 22398861312t^{15} + 104746549248t^{14} + 80245555200t^{13} +
1148205072384t^{12} - 320982220800t^{11} + 1675944787968t^{10} +
1433527123968t^{9} - 4728418467840t^{8} + 12606001643520t^{7} -
15188921155584t^{6} + 11311484239872t^{5} - 14888931950592t^{4} +
19135891243008t^{3} - 19494655229952t^{2} + 6675184484352t + 766450335744\]
\[B(t) = -27515376t^{36} - 437913216t^{35} - 1898494848t^{34} -
12171216384t^{33} - 154282952448t^{32} - 1181314990080t^{31} -
5686781755392t^{30} - 16898445213696t^{29} - 34579768983552t^{28} -
53143377149952t^{27} - 50195712835584t^{26} - 200541976657920t^{25} -
751461386158080t^{24} - 7040564880998400t^{23} - 20068358820986880t^{22} -
59571855421341696t^{21} - 96668090797916160t^{20} - 133704509187686400t^{19} -
230698477556858880t^{18} + 534818036750745600t^{17} - 1546689452766658560t^{16}
+ 3812598746965868544t^{15} - 5137499858172641280t^{14} +
7209538438142361600t^{13} - 3077985837703495680t^{12} +
3285679745563361280t^{11} - 3289626236392833024t^{10} +
13931217459597017088t^{9} - 36259515841697021952t^{8} +
70877216353585987584t^{7} - 95408365855070748672t^{6} +
79276707010440069120t^{5} - 41415014693405196288t^{4} +
13068744080454844416t^{3} - 8153973283784491008t^{2} + 7523291764824735744t -
1890842240914292736\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 564x - 37744$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |