Invariants
| Level: | $24$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.143 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&4\\14&1\end{bmatrix}$, $\begin{bmatrix}13&20\\20&3\end{bmatrix}$, $\begin{bmatrix}23&18\\6&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.f.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1536$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 13 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^2\cdot5^4}\cdot\frac{(x+y)^{24}(7x^{4}-12x^{3}y+72x^{2}y^{2}-792xy^{3}+3492y^{4})^{3}(97x^{4}-132x^{3}y+72x^{2}y^{2}-72xy^{3}+252y^{4})^{3}}{(x+y)^{24}(x^{2}-6y^{2})^{4}(x^{2}-18xy+6y^{2})^{4}(3x^{2}-4xy+18y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-8.b.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-12.a.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-24.b.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-24.b.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.144.4-24.i.1.10 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.bg.1.1 | $24$ | $4$ | $4$ | $3$ |
| 120.240.8-120.f.1.14 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-120.cu.1.2 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-120.f.1.28 | $120$ | $10$ | $10$ | $15$ |
| 168.384.11-168.i.1.29 | $168$ | $8$ | $8$ | $11$ |