Invariants
| Level: | $24$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.0.59 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&20\\18&7\end{bmatrix}$, $\begin{bmatrix}17&12\\10&7\end{bmatrix}$, $\begin{bmatrix}23&19\\12&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $6144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3}\cdot\frac{(2x-y)^{12}(396x^{4}-576x^{3}y+348x^{2}y^{2}-96xy^{3}+11y^{4})^{3}}{(2x-y)^{12}(6x^{2}-y^{2})^{4}(6x^{2}-4xy+y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.6.0.d.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 12.6.0.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 24.6.0.a.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.36.2.ce.1 | $24$ | $3$ | $3$ | $2$ |
| 24.48.1.im.1 | $24$ | $4$ | $4$ | $1$ |
| 72.324.22.dc.1 | $72$ | $27$ | $27$ | $22$ |
| 120.60.4.bg.1 | $120$ | $5$ | $5$ | $4$ |
| 120.72.3.bee.1 | $120$ | $6$ | $6$ | $3$ |
| 120.120.7.ce.1 | $120$ | $10$ | $10$ | $7$ |
| 168.96.5.fg.1 | $168$ | $8$ | $8$ | $5$ |
| 168.252.16.ce.1 | $168$ | $21$ | $21$ | $16$ |
| 168.336.21.ce.1 | $168$ | $28$ | $28$ | $21$ |
| 264.144.9.ijg.1 | $264$ | $12$ | $12$ | $9$ |
| 312.168.11.bs.1 | $312$ | $14$ | $14$ | $11$ |