Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.588 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&13\\4&15\end{bmatrix}$, $\begin{bmatrix}19&23\\12&23\end{bmatrix}$, $\begin{bmatrix}21&23\\8&7\end{bmatrix}$, $\begin{bmatrix}23&3\\4&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.0.bh.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| 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 54 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3}\cdot\frac{(3x+y)^{24}(9x^{4}-36x^{3}y-18x^{2}y^{2}+12xy^{3}+y^{4})^{3}(9x^{4}+36x^{3}y-18x^{2}y^{2}-12xy^{3}+y^{4})^{3}}{y^{2}x^{2}(3x+y)^{24}(3x^{2}-y^{2})^{2}(3x^{2}+y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-8.n.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.g.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-12.g.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-24.ba.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-24.ba.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.