Invariants
| Level: | $32$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $2\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
| Rouse and Zureick-Brown (RZB) label: | X239a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.96.0.106 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}27&15\\18&27\end{bmatrix}$, $\begin{bmatrix}31&16\\6&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 32.48.0.c.1 for the level structure with $-I$) |
| Cyclic 32-isogeny field degree: | $16$ |
| Cyclic 32-torsion field degree: | $256$ |
| Full 32-torsion field degree: | $4096$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^5\,\frac{(x-y)^{48}(458752x^{16}+524288x^{15}y+917504x^{14}y^{2}-4587520x^{13}y^{3}+802816x^{12}y^{4}+8945664x^{11}y^{5}+401408x^{10}y^{6}-5857280x^{9}y^{7}+125440x^{8}y^{8}+1464320x^{7}y^{9}+25088x^{6}y^{10}-139776x^{5}y^{11}+3136x^{4}y^{12}+4480x^{3}y^{13}+224x^{2}y^{14}-32xy^{15}+7y^{16})^{3}}{(x-y)^{48}(4x^{2}+y^{2})^{8}(256x^{8}-1024x^{7}y-1792x^{6}y^{2}+1792x^{5}y^{3}+1120x^{4}y^{4}-448x^{3}y^{5}-112x^{2}y^{6}+16xy^{7}+y^{8})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.c.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 32.48.0-16.c.1.1 | $32$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 96.288.8-96.i.1.5 | $96$ | $3$ | $3$ | $8$ |
| 96.384.7-96.dg.1.1 | $96$ | $4$ | $4$ | $7$ |
| 160.480.16-160.i.1.1 | $160$ | $5$ | $5$ | $16$ |