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: | 24.96.0.1012 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&1\\8&23\end{bmatrix}$, $\begin{bmatrix}11&13\\20&23\end{bmatrix}$, $\begin{bmatrix}17&20\\16&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1086443 |
| Contains $-I$: | no $\quad$ (see 24.48.0.bi.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^{11}}{3\cdot5^2}\cdot\frac{(x+y)^{48}(73x^{8}+204x^{7}y-4416x^{6}y^{2}+11808x^{5}y^{3}-23940x^{4}y^{4}-17712x^{3}y^{5}+152064x^{2}y^{6}-137376xy^{7}+123768y^{8})^{3}(191x^{8}-1272x^{7}y+8448x^{6}y^{2}-5904x^{5}y^{3}-47880x^{4}y^{4}+141696x^{3}y^{5}-317952x^{2}y^{6}+88128xy^{7}+189216y^{8})^{3}}{(x+y)^{48}(x^{2}-6y^{2})^{2}(x^{2}-18xy+6y^{2})^{4}(3x^{2}-4xy+18y^{2})^{2}(13x^{4}-18x^{3}y+18x^{2}y^{2}-108xy^{3}+468y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.ba.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-8.ba.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bh.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.bh.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.2.11 | $24$ | $2$ | $2$ | $0$ | $0$ |
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.gg.2.2 | $24$ | $3$ | $3$ | $8$ |
| 24.384.7-24.eh.2.1 | $24$ | $4$ | $4$ | $7$ |
| 48.192.1-48.cf.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ch.1.4 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cn.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cp.2.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dl.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dn.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dt.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dv.1.2 | $48$ | $2$ | $2$ | $1$ |
| 120.480.16-120.eo.2.2 | $120$ | $5$ | $5$ | $16$ |
| 240.192.1-240.ll.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ln.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lt.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lv.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qj.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ql.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qr.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qt.1.4 | $240$ | $2$ | $2$ | $1$ |