Curve name | $X_{193}$ | |||||||||
Index | $48$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | Yes | |||||||||
Generating matrices | $ \left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | $\tilde{X}_{1}(2,8)$ | |||||||||
Chosen covering | $X_{96}$ | |||||||||
Curves that $X_{193}$ minimally covers | $X_{96}$, $X_{98}$, $X_{102}$ | |||||||||
Curves that minimally cover $X_{193}$ | $X_{456}$, $X_{465}$, $X_{470}$, $X_{475}$, $X_{507}$, $X_{508}$, $X_{509}$, $X_{510}$, $X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$ | |||||||||
Curves that minimally cover $X_{193}$ and have infinitely many rational points. | $X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$ | |||||||||
Model | \[\mathbb{P}^{1}, \mathbb{Q}(X_{193}) = \mathbb{Q}(f_{193}), f_{96} = \frac{f_{193}}{f_{193}^{2} - \frac{1}{4}}\] | |||||||||
Info about rational points | None | |||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy + y = x^3 - 129473x - 10527244$, with conductor $25410$ | |||||||||
Generic density of odd order reductions | $17/168$ |