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: | $\begin{bmatrix}55&60\\152&103\end{bmatrix}$, $\begin{bmatrix}87&40\\180&15\end{bmatrix}$, $\begin{bmatrix}89&112\\154&37\end{bmatrix}$, $\begin{bmatrix}183&4\\98&47\end{bmatrix}$, $\begin{bmatrix}207&112\\194&41\end{bmatrix}$ |
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.1.8 | $232$ | $2$ | $2$ | $0$ | $?$ |
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.b.1.10 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.c.1.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.e.1.2 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.f.1.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.h.1.7 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.i.1.7 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.k.1.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.k.2.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-8.l.1.3 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.l.2.11 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.o.2.11 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.p.2.4 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.s.1.12 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.t.1.14 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.w.1.14 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.x.1.12 | $232$ | $2$ | $2$ | $0$ |
232.96.1-8.i.2.5 | $232$ | $2$ | $2$ | $1$ |
232.96.1-8.k.2.3 | $232$ | $2$ | $2$ | $1$ |
232.96.1-8.m.2.2 | $232$ | $2$ | $2$ | $1$ |
232.96.1-8.n.1.5 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.be.2.1 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bf.2.1 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bi.2.1 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bj.2.5 | $232$ | $2$ | $2$ | $1$ |