Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H0 |
| Rouse and Zureick-Brown (RZB) label: | X236f |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.0.277 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&8\\8&7\end{bmatrix}$, $\begin{bmatrix}9&1\\0&9\end{bmatrix}$, $\begin{bmatrix}15&0\\0&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.D_8$ |
| Contains $-I$: | no $\quad$ (see 16.48.0.z.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{(x-y)^{48}(x^{16}-16x^{14}y^{2}-912x^{12}y^{4}+7744x^{10}y^{6}+42080x^{8}y^{8}+30976x^{6}y^{10}-14592x^{4}y^{12}-1024x^{2}y^{14}+256y^{16})^{3}}{y^{4}x^{4}(x-y)^{48}(x^{2}-2y^{2})^{16}(x^{2}+2y^{2})^{2}(x^{2}-4xy+2y^{2})(x^{2}+4xy+2y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.bb.2.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.1.10 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.h.1.10 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.h.1.12 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-8.bb.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.