Invariants
| Level: | $16$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse and Zureick-Brown (RZB) label: | X42c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.0.43 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&1\\2&9\end{bmatrix}$, $\begin{bmatrix}11&0\\8&11\end{bmatrix}$, $\begin{bmatrix}13&3\\8&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.g.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $64$ |
| Full 16-torsion field degree: | $1024$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 40 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^7\,\frac{(4x-y)^{12}(4096x^{4}+4096x^{3}y+128x^{2}y^{2}-64xy^{3}+y^{4})^{3}}{(4x-y)^{12}(64x^{2}+y^{2})^{4}(64x^{2}+16xy-y^{2})^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 16.48.0-16.a.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.48.0-16.a.1.4 | $16$ | $2$ | $2$ | $0$ |
| 16.48.0-16.b.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.48.0-16.b.1.4 | $16$ | $2$ | $2$ | $0$ |
| 48.48.0-48.a.1.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.a.1.3 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.b.1.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.b.1.3 | $48$ | $2$ | $2$ | $0$ |
| 48.72.2-24.y.1.7 | $48$ | $3$ | $3$ | $2$ |
| 48.96.1-24.dp.1.3 | $48$ | $4$ | $4$ | $1$ |
| 80.48.0-80.a.1.1 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.a.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.b.1.1 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.b.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.120.4-40.m.1.4 | $80$ | $5$ | $5$ | $4$ |
| 80.144.3-40.s.1.4 | $80$ | $6$ | $6$ | $3$ |
| 80.240.7-40.y.1.6 | $80$ | $10$ | $10$ | $7$ |
| 112.48.0-112.a.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.a.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.b.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.b.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.192.5-56.m.1.6 | $112$ | $8$ | $8$ | $5$ |
| 112.504.16-56.y.1.1 | $112$ | $21$ | $21$ | $16$ |
| 176.48.0-176.a.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.a.1.3 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.b.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.b.1.3 | $176$ | $2$ | $2$ | $0$ |
| 176.288.9-88.m.1.7 | $176$ | $12$ | $12$ | $9$ |
| 208.48.0-208.a.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.a.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.b.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.b.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.336.11-104.s.1.3 | $208$ | $14$ | $14$ | $11$ |
| 240.48.0-240.a.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.a.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.b.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.b.1.5 | $240$ | $2$ | $2$ | $0$ |
| 272.48.0-272.a.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.a.1.3 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.b.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.b.1.3 | $272$ | $2$ | $2$ | $0$ |
| 272.432.15-136.s.1.1 | $272$ | $18$ | $18$ | $15$ |
| 304.48.0-304.a.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.48.0-304.a.1.3 | $304$ | $2$ | $2$ | $0$ |
| 304.48.0-304.b.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.48.0-304.b.1.3 | $304$ | $2$ | $2$ | $0$ |
| 304.480.17-152.m.1.8 | $304$ | $20$ | $20$ | $17$ |