Curve name | $X_{94e}$ | |||||||||
Index | $48$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | No | |||||||||
Generating matrices | $ \left[ \begin{matrix} 5 & 5 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||
Chosen covering | $X_{94}$ | |||||||||
Curves that $X_{94e}$ minimally covers | ||||||||||
Curves that minimally cover $X_{94e}$ | ||||||||||
Curves that minimally cover $X_{94e}$ and have infinitely many rational points. | ||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 324t^{16} + 10368t^{15} + 117504t^{14} + 387072t^{13} - 2875392t^{12} - 28753920t^{11} - 65912832t^{10} + 199950336t^{9} + 1299677184t^{8} + 1599602688t^{7} - 4218421248t^{6} - 14722007040t^{5} - 11777605632t^{4} + 12683575296t^{3} + 30802968576t^{2} + 21743271936t + 5435817984\] \[B(t) = 31104t^{23} + 1430784t^{22} + 27841536t^{21} + 287981568t^{20} + 1530814464t^{19} + 1571733504t^{18} - 30125260800t^{17} - 170378919936t^{16} - 155062370304t^{15} + 1805201178624t^{14} + 6837579546624t^{13} - 54700636372992t^{11} - 115532875431936t^{10} + 79391933595648t^{9} + 697872056057856t^{8} + 987144545894400t^{7} - 412020507672576t^{6} - 3210350614806528t^{5} - 4831528970354688t^{4} - 3736827705950208t^{3} - 1536292621910016t^{2} - 267181325549568t\] | |||||||||
Info about rational points | ||||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 28028x - 4961152$, with conductor $7840$ | |||||||||
Generic density of odd order reductions | $149/896$ |