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 $4$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
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: | 8N0 |
Level structure
$\GL_2(\Z/232\Z)$-generators: | $\begin{bmatrix}39&104\\64&125\end{bmatrix}$, $\begin{bmatrix}79&116\\32&161\end{bmatrix}$, $\begin{bmatrix}111&180\\80&139\end{bmatrix}$, $\begin{bmatrix}113&216\\20&99\end{bmatrix}$, $\begin{bmatrix}151&144\\56&33\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.48.0.c.1 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, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{48}(x^{8}-4x^{7}y+4x^{6}y^{2}+28x^{5}y^{3}+6x^{4}y^{4}-28x^{3}y^{5}+4x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+4x^{7}y+4x^{6}y^{2}-28x^{5}y^{3}+6x^{4}y^{4}+28x^{3}y^{5}+4x^{2}y^{6}-4xy^{7}+y^{8})^{3}}{y^{4}x^{52}(x-y)^{4}(x+y)^{4}(x^{2}+y^{2})^{8}(x^{2}-2xy-y^{2})^{4}(x^{2}+2xy-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
232.48.0-4.b.1.4 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-4.b.1.5 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-8.e.1.4 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-8.e.1.5 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-8.e.1.12 | $232$ | $2$ | $2$ | $0$ | $?$ |
232.48.0-8.e.1.13 | $232$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
232.192.1-8.f.1.1 | $232$ | $2$ | $2$ | $1$ |
232.192.1-8.f.1.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-8.f.2.5 | $232$ | $2$ | $2$ | $1$ |
232.192.1-8.g.1.3 | $232$ | $2$ | $2$ | $1$ |
232.192.1-8.g.1.5 | $232$ | $2$ | $2$ | $1$ |
232.192.1-8.g.2.3 | $232$ | $2$ | $2$ | $1$ |
232.192.3-8.i.1.4 | $232$ | $2$ | $2$ | $3$ |
232.192.3-8.j.1.4 | $232$ | $2$ | $2$ | $3$ |
232.192.1-232.w.1.1 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.w.1.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.w.2.10 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.x.1.4 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.x.1.6 | $232$ | $2$ | $2$ | $1$ |
232.192.1-232.x.2.11 | $232$ | $2$ | $2$ | $1$ |
232.192.3-232.be.1.8 | $232$ | $2$ | $2$ | $3$ |
232.192.3-232.bf.1.6 | $232$ | $2$ | $2$ | $3$ |