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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
$\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}87&228\\76&181\end{bmatrix}$, $\begin{bmatrix}143&24\\18&211\end{bmatrix}$, $\begin{bmatrix}169&68\\74&27\end{bmatrix}$, $\begin{bmatrix}169&88\\40&159\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 232.48.0.p.2 for the level structure with $-I$) |
Cyclic 232-isogeny field degree: | $60$ |
Cyclic 232-torsion field degree: | $6720$ |
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.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
232.48.0-8.e.1.12 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.e.1.6 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.e.1.19 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.i.1.6 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-232.i.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.u.1.3 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.z.1.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bm.1.5 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bo.1.1 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bx.1.4 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.bz.1.8 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.cg.1.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.ch.1.2 | $232$ | $2$ | $2$ | $1$ |