Invariants
| Level: | $132$ | $\SL_2$-level: | $12$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/132\Z)$-generators: | $\begin{bmatrix}48&47\\13&53\end{bmatrix}$, $\begin{bmatrix}59&100\\120&49\end{bmatrix}$, $\begin{bmatrix}64&33\\33&127\end{bmatrix}$, $\begin{bmatrix}91&71\\21&116\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.8.0.a.1 for the level structure with $-I$) |
| Cyclic 132-isogeny field degree: | $72$ |
| Cyclic 132-torsion field degree: | $2880$ |
| Full 132-torsion field degree: | $3801600$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 224 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{8}(x^{2}+12y^{2})^{3}(x^{2}+108y^{2})}{y^{6}x^{10}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $4$ | $2$ | $0$ | $0$ |
| 44.4.0-2.a.1.1 | $44$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.4.0-2.a.1.1 | $44$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 132.32.0-12.a.1.3 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-12.a.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-12.a.2.1 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-12.a.2.2 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-132.a.1.5 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-132.a.1.7 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-132.a.2.2 | $132$ | $2$ | $2$ | $0$ |
| 132.32.0-132.a.2.4 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-6.a.1.9 | $132$ | $3$ | $3$ | $0$ |
| 132.48.1-6.a.1.3 | $132$ | $3$ | $3$ | $1$ |
| 132.64.1-12.a.1.2 | $132$ | $4$ | $4$ | $1$ |
| 132.192.7-66.a.1.4 | $132$ | $12$ | $12$ | $7$ |
| 264.32.0-24.a.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-24.a.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-24.a.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-24.a.2.8 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-264.a.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-264.a.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-264.a.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.32.0-264.a.2.14 | $264$ | $2$ | $2$ | $0$ |