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: | $\begin{bmatrix}28&191\\21&142\end{bmatrix}$, $\begin{bmatrix}142&43\\231&26\end{bmatrix}$, $\begin{bmatrix}160&197\\257&224\end{bmatrix}$, $\begin{bmatrix}219&130\\88&197\end{bmatrix}$, $\begin{bmatrix}250&97\\43&60\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.ba.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 1542 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot3}\cdot\frac{(3x+y)^{12}(9x^{4}-192x^{2}y^{2}+256y^{4})^{3}}{y^{8}x^{2}(3x+y)^{12}(3x^{2}-64y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.12.0-4.c.1.2 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 264.12.0-4.c.1.5 | $264$ | $2$ | $2$ | $0$ | $?$ |
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.m.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.n.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bc.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.be.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bh.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bi.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bs.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bv.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.co.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cq.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cs.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cu.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dm.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.do.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.du.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dw.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-24.cu.1.3 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-24.iu.1.4 | $264$ | $4$ | $4$ | $1$ |
| 264.288.9-264.ikc.1.57 | $264$ | $12$ | $12$ | $9$ |