Invariants
| Level: | $24$ | $\SL_2$-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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.0.338 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&10\\8&9\end{bmatrix}$, $\begin{bmatrix}11&0\\16&1\end{bmatrix}$, $\begin{bmatrix}11&8\\4&15\end{bmatrix}$, $\begin{bmatrix}23&2\\20&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.r.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $768$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^4}\cdot\frac{(4x-y)^{48}(331776x^{8}-331776x^{7}y+110592x^{6}y^{2}-13824x^{5}y^{3}-1152x^{4}y^{4}+576x^{3}y^{5}+192x^{2}y^{6}+24xy^{7}+y^{8})^{3}(331776x^{8}+331776x^{7}y+110592x^{6}y^{2}+13824x^{5}y^{3}-1152x^{4}y^{4}-576x^{3}y^{5}+192x^{2}y^{6}-24xy^{7}+y^{8})^{3}}{y^{8}x^{8}(4x-y)^{48}(24x^{2}-y^{2})^{8}(24x^{2}+y^{2})^{4}(576x^{4}-144x^{2}y^{2}+y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.e.2.4 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.h.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.h.1.18 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.l.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.l.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.