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 $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | 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: | 8I0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}46&119\\181&40\end{bmatrix}$, $\begin{bmatrix}65&154\\66&217\end{bmatrix}$, $\begin{bmatrix}77&248\\218&75\end{bmatrix}$, $\begin{bmatrix}162&127\\41&104\end{bmatrix}$, $\begin{bmatrix}225&20\\40&77\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.2 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, including 131 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^4}\cdot\frac{(6x-y)^{24}(78941952x^{8}-342641664x^{7}y+528705792x^{6}y^{2}-425440512x^{5}y^{3}+202292640x^{4}y^{4}-58888512x^{3}y^{5}+10248336x^{2}y^{6}-966192xy^{7}+37199y^{8})^{3}}{(2x-y)^{4}(6x-y)^{26}(36x^{2}-24xy+5y^{2})(108x^{2}-60xy+11y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 264.24.0-8.n.1.3 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.24.0-8.n.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.j.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.m.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.n.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.o.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bg.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bi.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bi.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bk.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bk.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bm.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bm.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bo.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dz.2.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ed.2.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.eh.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.el.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.144.4-24.ge.2.22 | $264$ | $3$ | $3$ | $4$ |
| 264.192.3-24.gf.1.21 | $264$ | $4$ | $4$ | $3$ |