Invariants
| Level: | $176$ | $\SL_2$-level: | $16$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/176\Z)$-generators: | $\begin{bmatrix}36&43\\43&172\end{bmatrix}$, $\begin{bmatrix}57&80\\118&11\end{bmatrix}$, $\begin{bmatrix}83&76\\90&157\end{bmatrix}$, $\begin{bmatrix}85&158\\48&11\end{bmatrix}$, $\begin{bmatrix}120&79\\63&40\end{bmatrix}$, $\begin{bmatrix}145&32\\42&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 176-isogeny field degree: | $24$ |
| Cyclic 176-torsion field degree: | $1920$ |
| Full 176-torsion field degree: | $13516800$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 176.48.0-16.e.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.e.1.15 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.e.2.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.e.2.15 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.1.10 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.1.31 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.2.12 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.e.2.29 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.1.14 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.2.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.f.2.15 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.1.12 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.1.29 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.2.6 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.f.2.31 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.g.1.3 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.g.1.13 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.g.1.13 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.g.1.24 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.h.1.7 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-16.h.1.9 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.h.1.9 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.h.1.32 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.i.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.k.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.q.1.5 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.r.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.ba.1.6 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.ba.1.7 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.ba.2.2 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.ba.2.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.bb.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.bb.1.6 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.bb.2.3 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-8.bb.2.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bf.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bh.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bj.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bl.1.2 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bu.1.6 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bu.1.15 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bu.2.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bu.2.13 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bv.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bv.1.13 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bv.2.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-88.bv.2.15 | $176$ | $2$ | $2$ | $0$ |
| 176.48.1-16.a.1.7 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-16.a.1.9 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.a.1.9 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.a.1.32 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-16.b.1.3 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-16.b.1.13 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.b.1.13 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.b.1.24 | $176$ | $2$ | $2$ | $1$ |
| 176.288.9-88.bl.1.20 | $176$ | $12$ | $12$ | $9$ |