## The modular curve $X_{228f}$

Curve name $X_{228f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{84n}$
Meaning/Special name
Chosen covering $X_{228}$
Curves that $X_{228f}$ minimally covers
Curves that minimally cover $X_{228f}$
Curves that minimally cover $X_{228f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = 1944t^{24} + 134784t^{23} + 4202496t^{22} + 79045632t^{21} + 1003815936t^{20} + 9035587584t^{19} + 58033373184t^{18} + 251602993152t^{17} + 535696146432t^{16} - 1611125489664t^{15} - 21095843364864t^{14} - 108621912342528t^{13} - 367214645477376t^{12} - 868975298740224t^{11} - 1350133975351296t^{10} - 824896250707968t^{9} + 2194211415785472t^{8} + 8244526879604736t^{7} + 15213100579946496t^{6} + 18949000572960768t^{5} + 16841236782514176t^{4} + 10609325135364096t^{3} + 4512395720392704t^{2} + 1157785744048128t + 133590662774784$ $B(t) = -40824t^{36} - 2985984t^{35} - 97635456t^{34} - 1757528064t^{33} - 14342303232t^{32} + 127874433024t^{31} + 6166845259776t^{30} + 110870429958144t^{29} + 1352493952008192t^{28} + 12560963400105984t^{27} + 92759349685911552t^{26} + 554683331417997312t^{25} + 2694984873541632000t^{24} + 10517681266659164160t^{23} + 31776679396468850688t^{22} + 66603386247359496192t^{21} + 52945201293019840512t^{20} - 241599423328592855040t^{19} - 1152356389556116783104t^{18} - 1932795386628742840320t^{17} + 3388492882753269792768t^{16} + 34100933758648062050304t^{15} + 130157278807936412418048t^{14} + 344643379745887491194880t^{13} + 706474114689697579008000t^{12} + 1163255257849915898855424t^{11} + 1556243645700070264799232t^{10} + 1685903969053380131684352t^{9} + 1452229322978244541022208t^{8} + 952369741527374257717248t^{7} + 423782379363688708571136t^{6} + 70299713002575665037312t^{5} - 63078116690693223088128t^{4} - 61837521360340540981248t^{3} - 27481937703734278619136t^{2} - 6723838214867131564032t - 735419804751092514816$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 24x - 56$, with conductor $576$
Generic density of odd order reductions $109/896$