Invariants
Level: | $232$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 8C0 |
Level structure
$\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}44&185\\195&158\end{bmatrix}$, $\begin{bmatrix}135&112\\62&1\end{bmatrix}$, $\begin{bmatrix}177&164\\180&221\end{bmatrix}$, $\begin{bmatrix}197&104\\18&227\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.12.0.p.1 for the level structure with $-I$) |
Cyclic 232-isogeny field degree: | $60$ |
Cyclic 232-torsion field degree: | $6720$ |
Full 232-torsion field degree: | $43653120$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1543 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{12}(x^{2}-4xy-8y^{2})^{3}(x^{2}+4xy-8y^{2})^{3}}{y^{8}x^{14}(x^{2}-32y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
232.12.0-4.c.1.1 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.12.0-4.c.1.3 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
232.48.0-8.h.1.10 | $232$ | $2$ | $2$ | $0$ |
232.48.0-8.k.1.3 | $232$ | $2$ | $2$ | $0$ |
232.48.0-8.x.1.2 | $232$ | $2$ | $2$ | $0$ |
232.48.0-8.y.1.1 | $232$ | $2$ | $2$ | $0$ |
232.48.0-232.bt.1.7 | $232$ | $2$ | $2$ | $0$ |
232.48.0-232.bv.1.3 | $232$ | $2$ | $2$ | $0$ |
232.48.0-232.bx.1.5 | $232$ | $2$ | $2$ | $0$ |
232.48.0-232.bz.1.1 | $232$ | $2$ | $2$ | $0$ |