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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 8J0 |
Level structure
| $\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}9&108\\10&143\end{bmatrix}$, $\begin{bmatrix}23&148\\84&157\end{bmatrix}$, $\begin{bmatrix}97&96\\58&63\end{bmatrix}$, $\begin{bmatrix}145&140\\92&207\end{bmatrix}$, $\begin{bmatrix}167&120\\210&207\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.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 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 232.24.0-4.b.1.3 | $232$ | $2$ | $2$ | $0$ | $?$ |
| 232.24.0-4.b.1.4 | $232$ | $2$ | $2$ | $0$ | $?$ |
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.b.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.c.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.e.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.f.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.h.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.i.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.k.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.k.2.8 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.l.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.l.2.14 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.o.2.16 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.p.2.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.s.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.t.1.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.w.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.x.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-8.i.2.4 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.k.2.6 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.m.2.7 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.n.1.4 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.be.2.16 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bf.2.6 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bi.2.4 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bj.2.16 | $232$ | $2$ | $2$ | $1$ |