Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.0.1128 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&14\\16&23\end{bmatrix}$, $\begin{bmatrix}13&8\\12&19\end{bmatrix}$, $\begin{bmatrix}23&1\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1086683 |
| Contains $-I$: | no $\quad$ (see 24.48.0.bo.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3\cdot5^4}\cdot\frac{(2x+y)^{48}(6128x^{8}-71808x^{7}y-462624x^{6}y^{2}-753984x^{5}y^{3}-1194840x^{4}y^{4}-1130976x^{3}y^{5}-1040904x^{2}y^{6}-242352xy^{7}+31023y^{8})^{3}(60688x^{8}-739968x^{7}y+3268896x^{6}y^{2}-7769664x^{5}y^{3}+11718360x^{4}y^{4}-11654496x^{3}y^{5}+7355016x^{2}y^{6}-2497392xy^{7}+307233y^{8})^{3}}{(2x+y)^{48}(2x^{2}-3y^{2})^{4}(2x^{2}-18xy+3y^{2})^{2}(6x^{2}-4xy+9y^{2})^{2}(116x^{4}-288x^{3}y+1044x^{2}y^{2}-432xy^{3}+261y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.ba.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.ba.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bn.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bn.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.288.8-24.gs.2.3 | $24$ | $3$ | $3$ | $8$ |
| 24.384.7-24.er.2.5 | $24$ | $4$ | $4$ | $7$ |
| 48.192.1-48.cz.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.db.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dh.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dj.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ef.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.eh.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.en.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ep.1.2 | $48$ | $2$ | $2$ | $1$ |
| 120.480.16-120.fa.2.1 | $120$ | $5$ | $5$ | $16$ |
| 240.192.1-240.nh.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nj.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.np.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nr.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sf.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sh.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sn.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sp.1.4 | $240$ | $2$ | $2$ | $1$ |