Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 12E0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}111&136\\98&187\end{bmatrix}$, $\begin{bmatrix}168&263\\125&246\end{bmatrix}$, $\begin{bmatrix}178&143\\49&228\end{bmatrix}$, $\begin{bmatrix}238&243\\243&142\end{bmatrix}$, $\begin{bmatrix}250&69\\181&128\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.24.0.fn.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $48$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $20275200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 264.24.0-6.a.1.7 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.96.1-264.dj.1.38 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.gj.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.kc.1.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ke.1.12 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bku.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bkw.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bld.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.blf.1.12 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byi.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byk.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byr.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byt.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzs.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzu.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cab.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cad.1.24 | $264$ | $2$ | $2$ | $1$ |
| 264.144.1-264.bw.1.13 | $264$ | $3$ | $3$ | $1$ |