Invariants
| Level: | $264$ | $\SL_2$-level: | $8$ | ||||
| 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 $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}95&16\\100&99\end{bmatrix}$, $\begin{bmatrix}111&244\\164&175\end{bmatrix}$, $\begin{bmatrix}163&36\\4&43\end{bmatrix}$, $\begin{bmatrix}169&0\\128&235\end{bmatrix}$, $\begin{bmatrix}219&104\\220&147\end{bmatrix}$, $\begin{bmatrix}233&220\\232&171\end{bmatrix}$, $\begin{bmatrix}237&100\\92&257\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.b.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $7680$ |
| 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, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 264.24.0-4.b.1.4 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.24.0-4.b.1.9 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.96.0-8.a.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.a.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.a.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.a.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.a.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.a.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.a.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.a.1.35 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.b.2.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.1.20 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.1.24 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.2.22 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.1.22 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.2.18 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.b.2.21 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.31 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.33 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.43 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.27 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.32 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.42 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.c.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.c.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.c.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.c.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.c.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.c.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.c.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.c.1.28 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-8.g.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.g.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.g.1.12 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.g.2.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.h.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.h.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.h.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.h.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.2.17 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.n.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.n.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.n.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.n.2.17 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.1.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.24 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.37 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.2.11 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.2.14 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.2.20 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.o.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.o.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.o.2.12 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.o.2.21 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.14 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.28 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.37 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-8.a.1.7 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-8.a.1.8 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.a.1.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.a.1.9 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-88.a.1.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-88.a.1.9 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.a.1.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.a.1.23 | $264$ | $2$ | $2$ | $2$ |
| 264.144.4-12.b.1.35 | $264$ | $3$ | $3$ | $4$ |
| 264.192.3-12.b.1.23 | $264$ | $4$ | $4$ | $3$ |