Curve name | $X_{295}$ | ||||||||||||
Index | $48$ | ||||||||||||
Level | $16$ | ||||||||||||
Genus | $1$ | ||||||||||||
Does the subgroup contain $-I$? | Yes | ||||||||||||
Generating matrices | $ \left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 6 & 3 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{93}$ | ||||||||||||
Curves that $X_{295}$ minimally covers | $X_{93}$, $X_{104}$, $X_{167}$ | ||||||||||||
Curves that minimally cover $X_{295}$ | $X_{563}$, $X_{564}$, $X_{565}$, $X_{566}$ | ||||||||||||
Curves that minimally cover $X_{295}$ and have infinitely many rational points. | |||||||||||||
Model | \[y^2 = x^3 + x^2 - 9x + 7\] | ||||||||||||
Info about rational points | $X_{295}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ | ||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 + x^2 - 6627447363x - 207668330824122$, with conductor $1993828200$ | ||||||||||||
Generic density of odd order reductions | $2193/7168$ |