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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}1&156\\246&199\end{bmatrix}$, $\begin{bmatrix}83&104\\262&209\end{bmatrix}$, $\begin{bmatrix}97&172\\248&189\end{bmatrix}$, $\begin{bmatrix}215&236\\80&69\end{bmatrix}$, $\begin{bmatrix}239&240\\44&127\end{bmatrix}$, $\begin{bmatrix}243&212\\38&215\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 88.24.0.i.2 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 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 |
|---|---|---|---|---|---|
| 12.24.0-4.b.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 264.24.0-4.b.1.2 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.96.0-88.b.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.c.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.e.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.f.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.h.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.i.2.29 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.j.2.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.j.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.l.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.m.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.n.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.n.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.p.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.r.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.t.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.v.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.x.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.y.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.ba.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ba.2.29 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bb.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bd.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bi.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bl.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bq.2.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bt.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.by.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cb.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cn.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.co.2.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cr.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cs.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-88.q.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.s.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.x.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.y.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bd.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bf.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bh.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bj.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cc.1.12 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cd.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cg.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ch.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dv.2.20 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dy.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ed.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.eg.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.144.4-264.bp.1.18 | $264$ | $3$ | $3$ | $4$ |
| 264.192.3-264.dy.1.65 | $264$ | $4$ | $4$ | $3$ |