Invariants
| Level: | $232$ | $\SL_2$-level: | $8$ | ||||
| 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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}63&186\\60&205\end{bmatrix}$, $\begin{bmatrix}157&46\\12&21\end{bmatrix}$, $\begin{bmatrix}163&158\\52&85\end{bmatrix}$, $\begin{bmatrix}185&76\\132&191\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 232.48.0.o.2 for the level structure with $-I$) |
| Cyclic 232-isogeny field degree: | $60$ |
| Cyclic 232-torsion field degree: | $6720$ |
| Full 232-torsion field degree: | $10913280$ |
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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 232.48.0-8.e.1.12 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.48.0-232.e.1.13 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.48.0-232.e.1.19 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.48.0-232.h.2.16 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.48.0-232.h.2.28 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 232.192.1-232.j.2.5 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.z.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.bk.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.bo.2.5 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.bv.1.2 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.bz.2.7 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.cf.2.7 | $232$ | $2$ | $2$ | $1$ |
| 232.192.1-232.ch.1.2 | $232$ | $2$ | $2$ | $1$ |