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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}7&220\\48&35\end{bmatrix}$, $\begin{bmatrix}127&20\\132&163\end{bmatrix}$, $\begin{bmatrix}151&0\\174&113\end{bmatrix}$, $\begin{bmatrix}151&104\\96&167\end{bmatrix}$, $\begin{bmatrix}169&184\\6&169\end{bmatrix}$, $\begin{bmatrix}215&32\\242&157\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.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 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 264.24.0-4.b.1.3 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.24.0-4.b.1.4 | $264$ | $2$ | $2$ | $0$ | $?$ |
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.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.c.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.e.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.f.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.h.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.i.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.i.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.i.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.j.2.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.j.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.k.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-8.l.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.m.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.m.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.n.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.n.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.q.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.r.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.r.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.s.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.u.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.v.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.v.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.w.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bc.2.12 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.be.2.26 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bk.2.30 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bm.2.24 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bs.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bu.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ca.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cc.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-8.i.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.k.2.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.m.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-8.n.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.be.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.be.2.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.bf.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bf.2.14 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.bi.2.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bi.2.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.bj.2.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bj.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dx.2.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dz.2.32 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ef.2.32 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.eh.2.22 | $264$ | $2$ | $2$ | $1$ |
| 264.144.4-24.z.2.21 | $264$ | $3$ | $3$ | $4$ |
| 264.192.3-24.bq.2.45 | $264$ | $4$ | $4$ | $3$ |