| Curve name |
$X_{190b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{190}$ |
| Curves that $X_{190b}$ minimally covers |
|
| Curves that minimally cover $X_{190b}$ |
|
| Curves that minimally cover $X_{190b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3159t^{24} - 160704t^{23} - 4068576t^{22} - 68470272t^{21} -
863657856t^{20} - 8705470464t^{19} - 72648382464t^{18} - 510826217472t^{17} -
3046704648192t^{16} - 15428677533696t^{15} - 66259485720576t^{14} -
240950446129152t^{13} - 741147367440384t^{12} - 1927603569033216t^{11} -
4240607086116864t^{10} - 7899482897252352t^{9} - 12479302238994432t^{8} -
16738753494122496t^{7} - 19044337572642816t^{6} - 18256694794518528t^{5} -
14489774400208896t^{4} - 9189924343382016t^{3} - 4368600215322624t^{2} -
1380436848672768t - 217084827009024\]
\[B(t) = -51030t^{36} - 3312576t^{35} - 97689888t^{34} - 1600086528t^{33} -
11347019136t^{32} + 137995591680t^{31} + 5669491802112t^{30} +
100619809652736t^{29} + 1262385912840192t^{28} + 12491370505175040t^{27} +
102183265651654656t^{26} + 709551419585200128t^{25} + 4255210213459623936t^{24}
+ 22314137912271175680t^{23} + 103293962741474131968t^{22} +
425241710393699598336t^{21} + 1565939322271958040576t^{20} +
5179933655438123335680t^{19} + 15431571361347888218112t^{18} +
41439469243504986685440t^{17} + 100220116625405314596864t^{16} +
217723755721574194348032t^{15} + 423092071389078044540928t^{14} +
731189671109301884682240t^{13} + 1115477826197159657078784t^{12} +
1488037178685941618835456t^{11} + 1714350719423190921117696t^{10} +
1676563368810806111109120t^{9} + 1355476552644932778590208t^{8} +
864317583576492473843712t^{7} + 389604510000178299666432t^{6} +
75863878816994521251840t^{5} - 49904717922515124486144t^{4} -
56298039775479302455296t^{3} - 27497258949666593046528t^{2} -
7459258019618224078848t - 919274755938865643520\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 39x - 70$, with conductor $72$ |
| Generic density of odd order reductions |
$299/2688$ |