| Curve name |
$X_{207d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207d}$ minimally covers |
|
| Curves that minimally cover $X_{207d}$ |
|
| Curves that minimally cover $X_{207d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{30} - 6480t^{29} - 74736t^{28} + 2830464t^{27} + 111428352t^{26}
+ 1831624704t^{25} + 17489184768t^{24} + 98262319104t^{23} + 227672211456t^{22}
- 1027495231488t^{21} - 11106315730944t^{20} - 40916718452736t^{19} -
33829898158080t^{18} + 335189750317056t^{17} + 1579216105635840t^{16} +
2681518002536448t^{15} - 2165113482117120t^{14} - 20949359847800832t^{13} -
45491469233946624t^{12} - 33668963745398784t^{11} + 59682904199921664t^{10} +
206071019033591808t^{9} + 293419830516645888t^{8} + 245836506319552512t^{7} +
119645281921794048t^{6} + 24313500625010688t^{5} - 5135818813341696t^{4} -
3562417673994240t^{3} - 474989023199232t^{2}\]
\[B(t) = 432t^{45} + 38880t^{44} + 2524608t^{43} + 119035008t^{42} +
3749856768t^{41} + 79222800384t^{40} + 1133704581120t^{39} +
10574164942848t^{38} + 50047313412096t^{37} - 180547243868160t^{36} -
5547197342416896t^{35} - 51541099444961280t^{34} - 274005871363620864t^{33} -
699224096193380352t^{32} + 1539397909034827776t^{31} +
22876765719757848576t^{30} + 98016765829164564480t^{29} +
146610969832203485184t^{28} - 607190132200411496448t^{27} -
4235795678515314032640t^{26} - 10416869423878735134720t^{25} +
83334955391029881077760t^{23} + 271090923424980098088960t^{22} +
310881347686610686181376t^{21} - 600518532432705475313664t^{20} -
3211813382690064448880640t^{19} - 5997006872840201457106944t^{18} -
3228351403728207140093952t^{17} + 11731033694241119935660032t^{16} +
36776445513085454219476992t^{15} + 55341834128998112396574720t^{14} +
47650062340277363835666432t^{13} + 12407112124746939771125760t^{12} -
27513801517774654823989248t^{11} - 46505669234730873957384192t^{10} -
39888683820939501990051840t^{9} - 22299235893039349213691904t^{8} -
8443926771528766234558464t^{7} - 2144344070691376041295872t^{6} -
363834356737810341298176t^{5} - 44825588099114210426880t^{4} -
3984496719921263149056t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 35843892x + 3741877063568$, with conductor $141120$ |
| Generic density of odd order reductions |
$139/1344$ |