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 | $4^{8}\cdot8^{2}$ | 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 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.0.807 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&10\\20&5\end{bmatrix}$, $\begin{bmatrix}7&0\\4&13\end{bmatrix}$, $\begin{bmatrix}15&14\\4&11\end{bmatrix}$, $\begin{bmatrix}19&18\\4&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.n.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| 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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2}\cdot\frac{x^{48}(25458241x^{16}+230810336x^{15}y+1017129120x^{14}y^{2}+2743807360x^{13}y^{3}+5152387520x^{12}y^{4}+7191141888x^{11}y^{5}+7662151168x^{10}y^{6}+6317004800x^{9}y^{7}+4112745984x^{8}y^{8}+2129764352x^{7}y^{9}+881090560x^{6}y^{10}+267681792x^{5}y^{11}+62504960x^{4}y^{12}+10092544x^{3}y^{13}+2752512x^{2}y^{14}+524288xy^{15}+65536y^{16})^{3}}{x^{52}(x+2y)^{4}(5x^{2}-4xy-4y^{2})^{8}(7x^{2}+4xy+4y^{2})^{4}(73x^{4}+152x^{3}y+168x^{2}y^{2}+32xy^{3}+16y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.e.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.e.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.e.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.i.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.i.2.18 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.