Invariants
| Level: | $176$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/176\Z)$-generators: | $\begin{bmatrix}1&96\\46&117\end{bmatrix}$, $\begin{bmatrix}3&144\\47&121\end{bmatrix}$, $\begin{bmatrix}9&152\\153&35\end{bmatrix}$, $\begin{bmatrix}55&104\\40&55\end{bmatrix}$, $\begin{bmatrix}103&40\\86&95\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 88.48.0.bd.2 for the level structure with $-I$) |
| Cyclic 176-isogeny field degree: | $24$ |
| Cyclic 176-torsion field degree: | $960$ |
| Full 176-torsion field degree: | $3379200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.q.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 176.48.0-8.q.1.5 | $176$ | $2$ | $2$ | $0$ | $?$ |
| 176.48.0-88.bu.2.7 | $176$ | $2$ | $2$ | $0$ | $?$ |
| 176.48.0-88.bu.2.14 | $176$ | $2$ | $2$ | $0$ | $?$ |
| 176.48.0-88.bv.1.7 | $176$ | $2$ | $2$ | $0$ | $?$ |
| 176.48.0-88.bv.1.14 | $176$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 176.192.1-176.bf.1.5 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bg.2.5 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bh.2.5 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bk.2.3 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bl.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bo.2.5 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bp.2.3 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-176.bq.2.5 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-88.cj.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-88.ck.1.4 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-88.cl.2.4 | $176$ | $2$ | $2$ | $1$ |
| 176.192.1-88.cm.1.4 | $176$ | $2$ | $2$ | $1$ |
| 176.192.3-176.fq.2.13 | $176$ | $2$ | $2$ | $3$ |
| 176.192.3-176.fs.1.11 | $176$ | $2$ | $2$ | $3$ |
| 176.192.3-176.fy.2.13 | $176$ | $2$ | $2$ | $3$ |
| 176.192.3-176.gc.1.11 | $176$ | $2$ | $2$ | $3$ |