Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}159&104\\172&47\end{bmatrix}$, $\begin{bmatrix}173&144\\42&191\end{bmatrix}$, $\begin{bmatrix}177&64\\220&237\end{bmatrix}$, $\begin{bmatrix}183&164\\238&137\end{bmatrix}$, $\begin{bmatrix}204&173\\253&128\end{bmatrix}$, $\begin{bmatrix}224&123\\41&94\end{bmatrix}$, $\begin{bmatrix}225&238\\116&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $24$ |
| Cyclic 264-torsion field degree: | $1920$ |
| 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 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 88.12.0-4.c.1.1 | $88$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 88.12.0-4.c.1.1 | $88$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.96.0-12.c.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.3.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.3.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.4.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-12.c.4.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.1.46 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.2.43 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.3.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.3.43 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.4.10 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-132.c.4.37 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.1.28 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.2.24 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.1.28 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.2.24 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.3.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.3.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.4.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.4.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.1.50 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.2.27 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.2.38 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.1.27 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.1.38 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.2.51 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.1.31 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.1.33 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.2.29 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.2.35 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.3.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.3.59 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.4.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.4.53 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-12.b.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-12.h.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-12.k.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-132.k.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-12.l.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-132.l.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-132.o.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-132.p.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.cg.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.es.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ik.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.in.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iq.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iq.1.30 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ir.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ir.1.48 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.is.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.is.1.27 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.it.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.it.1.30 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iu.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iu.1.30 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iv.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iv.1.27 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iw.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iw.1.32 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ix.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ix.1.30 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.za.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zd.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zm.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zp.1.11 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zs.1.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zs.1.34 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zt.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zt.1.63 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zu.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zu.1.52 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zv.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zv.1.36 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zw.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zw.1.36 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zx.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zx.1.52 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zy.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zy.1.63 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zz.1.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zz.1.34 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-24.f.1.3 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.1.30 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.2.5 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.2.28 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.1.29 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.1.36 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.2.8 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.2.57 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.1.3 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.1.30 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.2.5 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.2.28 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.1.15 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.1.50 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.2.29 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.2.36 | $264$ | $2$ | $2$ | $2$ |
| 264.144.1-12.f.1.14 | $264$ | $3$ | $3$ | $1$ |