Invariants
| Level: | $132$ | $\SL_2$-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
| $\GL_2(\Z/132\Z)$-generators: | $\begin{bmatrix}23&16\\66&109\end{bmatrix}$, $\begin{bmatrix}72&59\\5&6\end{bmatrix}$, $\begin{bmatrix}108&29\\131&66\end{bmatrix}$, $\begin{bmatrix}113&0\\102&47\end{bmatrix}$, $\begin{bmatrix}121&86\\60&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 132-isogeny field degree: | $12$ |
| Cyclic 132-torsion field degree: | $480$ |
| Full 132-torsion field degree: | $1267200$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 44.12.0-4.c.1.2 | $44$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.12.0-4.c.1.2 | $44$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 132.96.0-12.c.1.2 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.1.7 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.2.2 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.2.7 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.3.2 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.3.7 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.4.2 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-12.c.4.7 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.1.2 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.1.15 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.2.4 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.2.13 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.3.8 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.3.9 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.4.7 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.c.4.10 | $132$ | $2$ | $2$ | $0$ |
| 132.96.1-12.b.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-12.h.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-12.k.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.k.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-12.l.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.l.1.5 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.o.1.3 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.p.1.7 | $132$ | $2$ | $2$ | $1$ |
| 132.144.1-12.f.1.4 | $132$ | $3$ | $3$ | $1$ |
| 264.96.0-24.bs.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.1.32 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bs.2.32 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.1.32 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bt.2.32 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.2.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.3.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.3.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.4.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.bu.4.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.1.31 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.1.34 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.do.2.59 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.1.31 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.1.34 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dp.2.55 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.1.29 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.1.35 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.2.25 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.2.39 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.3.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.3.57 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.4.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dq.4.49 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-24.cg.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.es.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ik.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.in.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iq.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iq.1.32 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ir.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ir.1.47 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.is.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.is.1.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.it.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.it.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iu.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iu.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iv.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iv.1.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iw.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.iw.1.31 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ix.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.ix.1.32 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.za.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zd.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zm.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zp.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zs.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zs.1.63 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zt.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zt.1.59 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zu.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zu.1.36 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zv.1.25 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zv.1.40 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zw.1.25 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zw.1.40 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zx.1.29 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zx.1.36 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zy.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zy.1.59 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zz.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zz.1.63 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-24.f.1.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.1.32 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.2.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.f.2.32 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.1.31 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.1.34 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.2.6 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.f.2.59 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.1.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.1.32 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.2.1 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.g.2.32 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.1.31 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.1.34 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.2.4 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.g.2.61 | $264$ | $2$ | $2$ | $2$ |