Invariants
| Level: | $176$ | $\SL_2$-level: | $16$ | ||||
| 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 | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 8G0 |
Level structure
| $\GL_2(\Z/176\Z)$-generators: | $\begin{bmatrix}19&12\\117&149\end{bmatrix}$, $\begin{bmatrix}21&172\\119&53\end{bmatrix}$, $\begin{bmatrix}79&36\\155&167\end{bmatrix}$, $\begin{bmatrix}145&148\\119&169\end{bmatrix}$, $\begin{bmatrix}159&20\\113&71\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.y.1 for the level structure with $-I$) |
| Cyclic 176-isogeny field degree: | $48$ |
| Cyclic 176-torsion field degree: | $3840$ |
| Full 176-torsion field degree: | $6758400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 15 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{(x-2y)^{24}(3x^{4}-48x^{3}y+272x^{2}y^{2}-640xy^{3}+704y^{4})^{3}(11x^{4}-80x^{3}y+272x^{2}y^{2}-384xy^{3}+192y^{4})^{3}}{(x-2y)^{24}(x^{2}-8y^{2})^{8}(x^{2}-8xy+8y^{2})^{2}(x^{2}-4xy+8y^{2})^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 176.96.0-16.r.1.5 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-16.r.1.7 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.r.1.9 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.r.1.14 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-16.s.1.6 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-16.s.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.s.1.11 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.s.1.16 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-16.o.1.3 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-16.o.1.7 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.o.1.9 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.o.1.12 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-16.p.1.5 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-16.p.1.7 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.p.1.9 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.p.1.12 | $176$ | $2$ | $2$ | $1$ |
| 176.96.2-16.i.1.4 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-16.i.1.8 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.j.1.6 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.j.1.16 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-16.o.1.2 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-16.o.1.6 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.o.1.2 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.o.1.12 | $176$ | $2$ | $2$ | $2$ |