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: | 24.96.0.521 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\8&17\end{bmatrix}$, $\begin{bmatrix}5&12\\0&11\end{bmatrix}$, $\begin{bmatrix}9&22\\8&7\end{bmatrix}$, $\begin{bmatrix}19&10\\20&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.v.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| 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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^4}\cdot\frac{(x-4y)^{48}(73x^{16}-4896x^{15}y+149376x^{14}y^{2}-2742912x^{13}y^{3}+34057152x^{12}y^{4}-304445952x^{11}y^{5}+2032846848x^{10}y^{6}-10360866816x^{9}y^{7}+40772408832x^{8}y^{8}-124330401792x^{7}y^{9}+292729946112x^{6}y^{10}-526082605056x^{5}y^{11}+706209103872x^{4}y^{12}-682524278784x^{3}y^{13}+446034345984x^{2}y^{14}-175432531968xy^{15}+31388663808y^{16})^{3}}{(x-6y)^{8}(x-4y)^{48}(x-2y)^{8}(x^{2}-12y^{2})^{8}(x^{2}-6xy+12y^{2})^{4}(x^{4}-24x^{3}y+144x^{2}y^{2}-288xy^{3}+144y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.e.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.h.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.h.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.m.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.m.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.