Curve name | $X_{58j}$ | |||||||||
Index | $48$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | No | |||||||||
Generating matrices | $ \left[ \begin{matrix} 7 & 4 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ | |||||||||
Images in lower levels |
|
|||||||||
Meaning/Special name | ||||||||||
Chosen covering | $X_{58}$ | |||||||||
Curves that $X_{58j}$ minimally covers | ||||||||||
Curves that minimally cover $X_{58j}$ | ||||||||||
Curves that minimally cover $X_{58j}$ and have infinitely many rational points. | ||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{14} + 54t^{12} - 405t^{10} + 756t^{8} - 405t^{6} + 54t^{4} - 27t^{2}\] \[B(t) = 54t^{21} - 162t^{19} - 1620t^{17} + 5292t^{15} - 7128t^{13} + 7128t^{11} - 5292t^{9} + 1620t^{7} + 162t^{5} - 54t^{3}\] | |||||||||
Info about rational points | ||||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 7488300x - 7225918000$, with conductor $187200$ | |||||||||
Generic density of odd order reductions | $25/224$ |