Curve name | $X_{76a}$ | |||||||||
Index | $48$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | No | |||||||||
Generating matrices | $ \left[ \begin{matrix} 5 & 5 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||
Chosen covering | $X_{76}$ | |||||||||
Curves that $X_{76a}$ minimally covers | ||||||||||
Curves that minimally cover $X_{76a}$ | ||||||||||
Curves that minimally cover $X_{76a}$ and have infinitely many rational points. | ||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -2430t^{20} + 47952t^{19} + 227664t^{18} - 482112t^{17} + 3227040t^{16} + 13353984t^{15} + 3898368t^{14} + 67350528t^{13} - 210263040t^{12} + 1371561984t^{11} - 1515331584t^{10} - 5486247936t^{9} - 3364208640t^{8} - 4310433792t^{7} + 997982208t^{6} - 13674479616t^{5} + 13217955840t^{4} + 7898923008t^{3} + 14920187904t^{2} - 12570329088t - 2548039680\] \[B(t) = -198288t^{30} - 933120t^{29} - 3810240t^{28} - 5197824t^{27} + 178972416t^{26} - 731566080t^{25} - 952777728t^{24} + 33570422784t^{23} - 32805015552t^{22} - 60436316160t^{21} + 1389027999744t^{20} - 91814363136t^{19} + 1315314008064t^{18} + 11691680071680t^{17} + 27920845504512t^{16} + 111683382018048t^{14} - 187066881146880t^{13} + 84180096516096t^{12} + 23504476962816t^{11} + 1422364671737856t^{10} + 247547150991360t^{9} - 537477374803968t^{8} - 2200071227572224t^{7} - 249764964728832t^{6} + 767102633902080t^{5} + 750664720318464t^{4} + 87205015977984t^{3} - 255700877967360t^{2} + 250482492702720t - 212910118797312\] | |||||||||
Info about rational points | ||||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 120x + 2176$, with conductor $2304$ | |||||||||
Generic density of odd order reductions | $403/1792$ |