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: | $\begin{bmatrix}25&24\\42&71\end{bmatrix}$, $\begin{bmatrix}75&200\\112&215\end{bmatrix}$, $\begin{bmatrix}119&120\\80&81\end{bmatrix}$, $\begin{bmatrix}143&88\\170&73\end{bmatrix}$, $\begin{bmatrix}195&132\\194&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.2 for the level structure with $-I$) |
| Cyclic 232-isogeny field degree: | $60$ |
| Cyclic 232-torsion field degree: | $3360$ |
| 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 222 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{x^{24}(x^{8}-32x^{6}y^{2}+1280x^{4}y^{4}-16384x^{2}y^{6}+65536y^{8})^{3}}{y^{4}x^{32}(x-4y)^{4}(x+4y)^{4}(x^{2}-8y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 116.24.0-4.b.1.1 | $116$ | $2$ | $2$ | $0$ | $?$ |
| 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 |
|---|---|---|---|---|
| 232.96.0-8.b.2.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.c.1.10 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.e.2.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.f.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.h.2.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.i.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.k.2.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.k.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-8.l.2.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.l.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.o.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.p.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.s.2.8 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.t.2.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.w.2.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.x.2.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-8.i.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.k.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.m.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-8.n.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.be.1.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bf.1.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bi.1.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bj.1.9 | $232$ | $2$ | $2$ | $1$ |