Invariants
| Level: | $264$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}39&98\\8&123\end{bmatrix}$, $\begin{bmatrix}103&16\\16&227\end{bmatrix}$, $\begin{bmatrix}111&164\\104&15\end{bmatrix}$, $\begin{bmatrix}115&100\\70&219\end{bmatrix}$, $\begin{bmatrix}127&76\\214&5\end{bmatrix}$, $\begin{bmatrix}139&212\\252&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 2.6.0.a.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $192$ |
| Cyclic 264-torsion field degree: | $15360$ |
| Full 264-torsion field degree: | $81100800$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 31720 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+192y^{2})^{3}}{y^{2}x^{6}(x-8y)^{2}(x+8y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 264.4.0-2.a.1.1 | $264$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.24.0-4.a.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-8.a.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-12.a.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-24.a.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-44.a.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-88.a.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-132.a.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.a.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-4.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-8.b.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-12.b.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-24.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-44.b.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-88.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-132.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.b.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.36.1-6.a.1.6 | $264$ | $3$ | $3$ | $1$ |
| 264.48.0-6.a.1.11 | $264$ | $4$ | $4$ | $0$ |
| 264.144.4-22.a.1.12 | $264$ | $12$ | $12$ | $4$ |