Invariants
| Level: | $232$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8J0 |
Level structure
| $\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}5&116\\214&229\end{bmatrix}$, $\begin{bmatrix}33&52\\158&185\end{bmatrix}$, $\begin{bmatrix}85&180\\158&9\end{bmatrix}$, $\begin{bmatrix}103&20\\8&225\end{bmatrix}$, $\begin{bmatrix}177&0\\36&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 232.24.0.h.2 for the level structure with $-I$) |
| Cyclic 232-isogeny field degree: | $60$ |
| Cyclic 232-torsion field degree: | $6720$ |
| Full 232-torsion field degree: | $21826560$ |
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 |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 232.24.0-4.b.1.8 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 232.96.0-232.a.1.13 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.b.1.13 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.d.1.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.e.1.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.i.2.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.k.1.16 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.m.1.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.o.2.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.q.2.16 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.s.1.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.u.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.w.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.y.1.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.z.2.16 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bb.2.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bc.2.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.m.1.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.q.1.16 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.w.1.8 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.x.1.6 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bc.2.16 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.be.1.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bg.1.6 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bi.2.8 | $232$ | $2$ | $2$ | $1$ |