Invariants
| Level: | $16$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse and Zureick-Brown (RZB) label: | X123b |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.0.240 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&9\\4&1\end{bmatrix}$, $\begin{bmatrix}5&3\\6&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.24.0.c.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $64$ |
| Full 16-torsion field degree: | $512$ |
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 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^5\,\frac{(x-y)^{24}(1792x^{8}+1024x^{7}y+1792x^{6}y^{2}-1792x^{5}y^{3}+672x^{4}y^{4}+448x^{3}y^{5}+112x^{2}y^{6}-16xy^{7}+7y^{8})^{3}}{(x-y)^{24}(4x^{2}+y^{2})^{4}(16x^{4}-32x^{3}y-24x^{2}y^{2}+8xy^{3}+y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.h.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 16.24.0-8.h.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 32.96.0-32.c.1.1 | $32$ | $2$ | $2$ | $0$ |
| 32.96.0-32.d.1.1 | $32$ | $2$ | $2$ | $0$ |
| 48.144.4-48.i.1.9 | $48$ | $3$ | $3$ | $4$ |
| 48.192.3-48.gv.1.1 | $48$ | $4$ | $4$ | $3$ |
| 80.240.8-80.e.1.1 | $80$ | $5$ | $5$ | $8$ |
| 80.288.7-80.k.1.1 | $80$ | $6$ | $6$ | $7$ |
| 80.480.15-80.i.1.2 | $80$ | $10$ | $10$ | $15$ |
| 96.96.0-96.c.1.1 | $96$ | $2$ | $2$ | $0$ |
| 96.96.0-96.d.1.1 | $96$ | $2$ | $2$ | $0$ |
| 112.384.11-112.g.1.1 | $112$ | $8$ | $8$ | $11$ |
| 160.96.0-160.g.1.1 | $160$ | $2$ | $2$ | $0$ |
| 160.96.0-160.h.1.1 | $160$ | $2$ | $2$ | $0$ |
| 224.96.0-224.c.1.1 | $224$ | $2$ | $2$ | $0$ |
| 224.96.0-224.d.1.1 | $224$ | $2$ | $2$ | $0$ |