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: | 24.96.0.331 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\20&23\end{bmatrix}$, $\begin{bmatrix}7&22\\12&11\end{bmatrix}$, $\begin{bmatrix}13&10\\8&9\end{bmatrix}$, $\begin{bmatrix}13&18\\8&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.w.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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{(x+y)^{48}(x^{16}+16x^{15}y+144x^{14}y^{2}+896x^{13}y^{3}+4112x^{12}y^{4}+14400x^{11}y^{5}+38944x^{10}y^{6}+81088x^{9}y^{7}+147312x^{8}y^{8}+302464x^{7}y^{9}+654208x^{6}y^{10}+1115136x^{5}y^{11}+1317632x^{4}y^{12}+1046528x^{3}y^{13}+562176x^{2}y^{14}+206848xy^{15}+49408y^{16})^{3}}{y^{8}(x+y)^{56}(x^{2}+2xy-2y^{2})^{4}(x^{2}+2xy+4y^{2})^{8}(x^{4}+4x^{3}y+24x^{2}y^{2}+40xy^{3}+28y^{4})^{2}}$ |
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.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.i.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.m.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.m.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.