Invariants
| Level: | $264$ | $\SL_2$-level: | $8$ | ||||
| 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 | $1^{2}\cdot2\cdot8$ | 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}43&118\\212&225\end{bmatrix}$, $\begin{bmatrix}179&234\\132&73\end{bmatrix}$, $\begin{bmatrix}196&81\\59&2\end{bmatrix}$, $\begin{bmatrix}242&69\\111&20\end{bmatrix}$, $\begin{bmatrix}247&68\\58&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.bb.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $7680$ |
| Full 264-torsion field degree: | $40550400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 950 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3}\cdot\frac{x^{12}(9x^{4}+384x^{2}y^{2}+1024y^{4})^{3}}{y^{8}x^{14}(3x^{2}+128y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 88.12.0-4.c.1.6 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 132.12.0-4.c.1.2 | $132$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.48.0-24.l.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.o.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bd.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.be.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bg.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bj.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bt.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bu.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cp.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cr.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ct.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cv.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dn.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dp.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dv.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dx.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-24.cv.1.21 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-24.iv.1.32 | $264$ | $4$ | $4$ | $1$ |
| 264.288.9-264.ikd.1.3 | $264$ | $12$ | $12$ | $9$ |