Curve name | $X_{203h}$ | |||||||||
Index | $96$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | No | |||||||||
Generating matrices | $ \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||
Chosen covering | $X_{203}$ | |||||||||
Curves that $X_{203h}$ minimally covers | ||||||||||
Curves that minimally cover $X_{203h}$ | ||||||||||
Curves that minimally cover $X_{203h}$ and have infinitely many rational points. | ||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -7421703487488t^{24} - 18554258718720t^{22} - 44762149158912t^{20} - 62214748766208t^{18} - 41252422680576t^{16} - 13634843443200t^{14} - 2457782452224t^{12} - 213044428800t^{10} - 10071392256t^{8} - 237330432t^{6} - 2668032t^{4} - 17280t^{2} - 108\] \[B(t) = -7782220156096217088t^{36} - 29183325585360814080t^{34} + 16415620641765457920t^{32} + 163183428898142552064t^{30} + 271610123201877639168t^{28} + 244940439527287160832t^{26} + 142280943195981348864t^{24} + 54465625696814235648t^{22} + 13305326650238435328t^{20} + 2060618114827026432t^{18} + 207895728909975552t^{16} + 13297271898636288t^{14} + 542758724960256t^{12} + 14599587889152t^{10} + 252956639232t^{8} + 2374631424t^{6} + 3732480t^{4} - 103680t^{2} - 432\] | |||||||||
Info about rational points | ||||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - x^2 - 32072449x + 69918983329$, with conductor $55488$ | |||||||||
Generic density of odd order reductions | $109/896$ |