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: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse and Zureick-Brown (RZB) label: | X40c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.0.26 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&5\\10&5\end{bmatrix}$, $\begin{bmatrix}3&1\\10&5\end{bmatrix}$, $\begin{bmatrix}9&13\\6&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.i.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 33 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(x-4y)^{12}(5x^{4}-32x^{3}y+64x^{2}y^{2}-1024xy^{3}+5120y^{4})^{3}}{(x-4y)^{12}(x^{2}-32y^{2})^{2}(x^{2}-16xy+32y^{2})^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 16.48.1-8.k.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1-8.k.1.9 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1-8.l.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1-8.l.1.7 | $16$ | $2$ | $2$ | $1$ |
| 48.48.1-24.k.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.k.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.l.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-24.l.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.72.2-24.ba.1.10 | $48$ | $3$ | $3$ | $2$ |
| 48.96.1-24.dr.1.7 | $48$ | $4$ | $4$ | $1$ |
| 80.48.1-40.k.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-40.k.1.4 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-40.l.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-40.l.1.4 | $80$ | $2$ | $2$ | $1$ |
| 80.120.4-40.o.1.5 | $80$ | $5$ | $5$ | $4$ |
| 80.144.3-40.u.1.13 | $80$ | $6$ | $6$ | $3$ |
| 80.240.7-40.ba.1.9 | $80$ | $10$ | $10$ | $7$ |
| 112.48.1-56.k.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-56.k.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-56.l.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-56.l.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.192.5-56.o.1.7 | $112$ | $8$ | $8$ | $5$ |
| 112.504.16-56.ba.1.15 | $112$ | $21$ | $21$ | $16$ |
| 176.48.1-88.k.1.6 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-88.k.1.8 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-88.l.1.6 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-88.l.1.8 | $176$ | $2$ | $2$ | $1$ |
| 176.288.9-88.o.1.6 | $176$ | $12$ | $12$ | $9$ |
| 208.48.1-104.k.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-104.k.1.7 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-104.l.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-104.l.1.7 | $208$ | $2$ | $2$ | $1$ |
| 208.336.11-104.u.1.14 | $208$ | $14$ | $14$ | $11$ |
| 240.48.1-120.k.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.k.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.l.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-120.l.1.13 | $240$ | $2$ | $2$ | $1$ |
| 272.48.1-136.k.1.5 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-136.k.1.8 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-136.l.1.5 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-136.l.1.8 | $272$ | $2$ | $2$ | $1$ |
| 272.432.15-136.u.1.15 | $272$ | $18$ | $18$ | $15$ |
| 304.48.1-152.k.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1-152.k.1.3 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1-152.l.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1-152.l.1.5 | $304$ | $2$ | $2$ | $1$ |
| 304.480.17-152.o.1.5 | $304$ | $20$ | $20$ | $17$ |