Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | ||||
| 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 | $3^{4}$ | 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 $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 3D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.3 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}2&9\\3&14\end{bmatrix}$, $\begin{bmatrix}4&21\\21&10\end{bmatrix}$, $\begin{bmatrix}16&15\\21&10\end{bmatrix}$, $\begin{bmatrix}22&9\\3&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 3.12.0.a.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $12$ |
| Cyclic 24-torsion field degree: | $96$ |
| 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 1550 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{15}(x+6y)^{3}(x^{2}-6xy+36y^{2})^{3}}{y^{3}x^{12}(x-3y)^{3}(x^{2}+3xy+9y^{2})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.8.0-3.a.1.7 | $24$ | $3$ | $3$ | $0$ | $0$ |
| 24.8.0-3.a.1.8 | $24$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.48.1-6.a.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-6.b.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.72.0-6.a.1.6 | $24$ | $3$ | $3$ | $0$ |
| 72.72.0-9.a.1.5 | $72$ | $3$ | $3$ | $0$ |
| 72.72.0-9.a.1.7 | $72$ | $3$ | $3$ | $0$ |
| 72.72.0-9.b.1.3 | $72$ | $3$ | $3$ | $0$ |
| 72.72.0-9.c.1.3 | $72$ | $3$ | $3$ | $0$ |
| 72.72.1-9.a.1.5 | $72$ | $3$ | $3$ | $1$ |
| 72.72.1-9.a.1.7 | $72$ | $3$ | $3$ | $1$ |
| 72.72.2-9.a.1.3 | $72$ | $3$ | $3$ | $2$ |
| 24.48.1-12.b.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-12.d.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.3-12.o.1.1 | $24$ | $4$ | $4$ | $3$ |
| 120.120.4-15.a.1.6 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-15.a.1.10 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-15.a.1.12 | $120$ | $10$ | $10$ | $7$ |
| 168.192.5-21.a.1.13 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-21.a.1.10 | $168$ | $21$ | $21$ | $16$ |
| 24.48.1-24.bx.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.cd.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.ci.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.cl.1.1 | $24$ | $2$ | $2$ | $1$ |
| 120.48.1-30.c.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-30.e.1.2 | $120$ | $2$ | $2$ | $1$ |
| 264.288.9-33.a.1.15 | $264$ | $12$ | $12$ | $9$ |
| 312.336.11-39.a.1.10 | $312$ | $14$ | $14$ | $11$ |
| 168.48.1-42.b.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-42.c.1.1 | $168$ | $2$ | $2$ | $1$ |
| 120.48.1-60.g.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.m.1.1 | $120$ | $2$ | $2$ | $1$ |
| 264.48.1-66.b.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-66.c.1.4 | $264$ | $2$ | $2$ | $1$ |
| 312.48.1-78.b.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-78.c.1.4 | $312$ | $2$ | $2$ | $1$ |
| 168.48.1-84.d.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-84.g.1.2 | $168$ | $2$ | $2$ | $1$ |
| 120.48.1-120.cu.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.cx.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.ey.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.fb.1.5 | $120$ | $2$ | $2$ | $1$ |
| 264.48.1-132.d.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.g.1.1 | $264$ | $2$ | $2$ | $1$ |
| 312.48.1-156.d.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-156.g.1.1 | $312$ | $2$ | $2$ | $1$ |
| 168.48.1-168.ci.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.cl.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.ea.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.ed.1.5 | $168$ | $2$ | $2$ | $1$ |
| 264.48.1-264.ci.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.cl.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.ea.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.ed.1.5 | $264$ | $2$ | $2$ | $1$ |
| 312.48.1-312.ci.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.cl.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.ea.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.ed.1.7 | $312$ | $2$ | $2$ | $1$ |