Invariants
| Level: | $264$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}113&112\\50&201\end{bmatrix}$, $\begin{bmatrix}193&236\\168&227\end{bmatrix}$, $\begin{bmatrix}195&104\\178&63\end{bmatrix}$, $\begin{bmatrix}245&168\\242&119\end{bmatrix}$, $\begin{bmatrix}249&220\\112&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.w.2 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $10137600$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{(x+y)^{48}(x^{16}+16x^{15}y+144x^{14}y^{2}+896x^{13}y^{3}+4112x^{12}y^{4}+14400x^{11}y^{5}+38944x^{10}y^{6}+81088x^{9}y^{7}+147312x^{8}y^{8}+302464x^{7}y^{9}+654208x^{6}y^{10}+1115136x^{5}y^{11}+1317632x^{4}y^{12}+1046528x^{3}y^{13}+562176x^{2}y^{14}+206848xy^{15}+49408y^{16})^{3}}{y^{8}(x+y)^{56}(x^{2}+2xy-2y^{2})^{4}(x^{2}+2xy+4y^{2})^{8}(x^{4}+4x^{3}y+24x^{2}y^{2}+40xy^{3}+28y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 88.48.0-8.e.2.10 | $88$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-8.e.2.16 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-24.i.2.19 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-24.i.2.28 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-24.m.1.15 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-24.m.1.18 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.192.1-24.s.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.t.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.x.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.y.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.bm.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.bn.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.bo.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-24.bp.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.288.8-24.er.1.30 | $264$ | $3$ | $3$ | $8$ |
| 264.384.7-24.cx.1.16 | $264$ | $4$ | $4$ | $7$ |
| 264.192.1-264.nq.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.nr.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ns.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.nt.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.og.2.3 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.oh.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.oi.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.oj.1.2 | $264$ | $2$ | $2$ | $1$ |