Curve name | $X_{242c}$ | |||||||||||||||
Index | $96$ | |||||||||||||||
Level | $32$ | |||||||||||||||
Genus | $0$ | |||||||||||||||
Does the subgroup contain $-I$? | No | |||||||||||||||
Generating matrices | $ \left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right]$ | |||||||||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||||||||
Chosen covering | $X_{242}$ | |||||||||||||||
Curves that $X_{242c}$ minimally covers | ||||||||||||||||
Curves that minimally cover $X_{242c}$ | ||||||||||||||||
Curves that minimally cover $X_{242c}$ and have infinitely many rational points. | ||||||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{16} - 1728t^{8} - 1728\] \[B(t) = 432t^{24} + 10368t^{16} + 51840t^{8} - 27648\] | |||||||||||||||
Info about rational points | ||||||||||||||||
Comments on finding rational points | None | |||||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 44x + 48$, with conductor $1088$ | |||||||||||||||
Generic density of odd order reductions | $13411/86016$ |