Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| 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: | 6F0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}1&98\\36&233\end{bmatrix}$, $\begin{bmatrix}16&191\\247&210\end{bmatrix}$, $\begin{bmatrix}99&40\\194&193\end{bmatrix}$, $\begin{bmatrix}142&71\\3&134\end{bmatrix}$, $\begin{bmatrix}247&210\\6&127\end{bmatrix}$, $\begin{bmatrix}254&187\\227&78\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $48$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $40550400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.48.0-6.a.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-66.a.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-6.b.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-66.b.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.d.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.f.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.g.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.h.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.i.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-12.j.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.m.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.n.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.o.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.p.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.p.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.q.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.r.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.y.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bw.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.bx.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.ca.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.cb.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.cc.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.cd.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fg.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fh.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fi.1.30 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fj.1.32 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fk.1.22 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fl.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fm.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fn.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-12.i.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-12.j.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-12.k.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-12.l.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.m.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.n.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.o.1.12 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.p.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-24.eq.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-24.er.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-24.es.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-24.et.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hk.1.21 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hl.1.27 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hm.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hn.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.72.0-6.a.1.3 | $264$ | $3$ | $3$ | $0$ |
| 264.288.9-66.a.1.48 | $264$ | $12$ | $12$ | $9$ |