Invariants
Level: | $232$ | $\SL_2$-level: | $8$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
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: | 8O0 |
Level structure
$\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}5&160\\64&113\end{bmatrix}$, $\begin{bmatrix}43&60\\126&27\end{bmatrix}$, $\begin{bmatrix}113&228\\16&165\end{bmatrix}$, $\begin{bmatrix}165&0\\200&207\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 232.48.0.s.2 for the level structure with $-I$) |
Cyclic 232-isogeny field degree: | $60$ |
Cyclic 232-torsion field degree: | $3360$ |
Full 232-torsion field degree: | $10913280$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
232.48.0-8.e.2.9 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.h.1.20 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.h.1.21 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.l.1.14 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.l.1.18 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
232.192.1-232.g.1.1 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.j.1.4 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.w.2.5 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.z.2.5 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bc.2.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bd.2.2 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bg.1.2 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bh.1.4 | $232$ | $2$ | $2$ | $1$ |