Invariants
| Level: | $24$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.271 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&11\\4&13\end{bmatrix}$, $\begin{bmatrix}5&1\\0&1\end{bmatrix}$, $\begin{bmatrix}9&16\\8&15\end{bmatrix}$, $\begin{bmatrix}17&20\\4&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.k.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $3072$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 775 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{(x-y)^{12}(x^{4}+28x^{2}y^{2}+4y^{4})^{3}}{y^{2}x^{2}(x-y)^{12}(x^{2}-2y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-4.c.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 24.12.0-4.c.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.48.0-8.r.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.r.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.w.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.w.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.y.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.y.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.be.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.be.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-24.bu.1.6 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-24.es.1.4 | $24$ | $4$ | $4$ | $1$ |
| 120.48.0-40.x.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.x.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bg.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bg.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bj.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bj.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bs.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bs.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.120.4-40.w.1.2 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-40.bi.1.8 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-40.bu.1.8 | $120$ | $10$ | $10$ | $7$ |
| 168.48.0-56.v.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.v.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bc.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bc.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bh.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bh.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bo.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bo.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.192.5-56.w.1.1 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-56.bu.1.6 | $168$ | $21$ | $21$ | $16$ |
| 264.48.0-88.v.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.v.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bc.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bc.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bh.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bh.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bo.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bo.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.288.9-88.w.1.3 | $264$ | $12$ | $12$ | $9$ |
| 312.48.0-104.x.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.x.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bg.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bg.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bj.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bj.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bs.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bs.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.336.11-104.bi.1.2 | $312$ | $14$ | $14$ | $11$ |