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: | $\begin{bmatrix}43&228\\0&167\end{bmatrix}$, $\begin{bmatrix}73&60\\6&27\end{bmatrix}$, $\begin{bmatrix}133&48\\84&219\end{bmatrix}$, $\begin{bmatrix}165&176\\30&89\end{bmatrix}$, $\begin{bmatrix}219&204\\166&71\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.d.1 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, including 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 232.24.0-4.b.1.7 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.24.0-4.b.1.9 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 232.96.0-8.a.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.b.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.d.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.e.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.g.1.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.h.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.i.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.j.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.j.2.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.k.2.8 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.m.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.n.2.11 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.q.1.12 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.r.2.12 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.u.2.12 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.v.1.12 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-8.e.2.4 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.i.1.6 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.l.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.m.2.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bc.2.3 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bd.2.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bg.2.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bh.2.3 | $232$ | $2$ | $2$ | $1$ |