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: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/132\Z)$-generators: | $\begin{bmatrix}24&97\\109&72\end{bmatrix}$, $\begin{bmatrix}27&38\\70&53\end{bmatrix}$, $\begin{bmatrix}60&71\\13&32\end{bmatrix}$, $\begin{bmatrix}94&23\\31&60\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 66.24.0.a.1 for the level structure with $-I$) |
| Cyclic 132-isogeny field degree: | $24$ |
| Cyclic 132-torsion field degree: | $960$ |
| 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 38 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{20}}{3^6\cdot11^3}\cdot\frac{(x+4y)^{24}(9x^{2}-30xy-8y^{2})^{3}(29889x^{6}+903960x^{5}y-291600x^{4}y^{2}-11232000x^{3}y^{3}+38465280x^{2}y^{4}-50227200xy^{5}+30543872y^{6})^{3}}{(x+4y)^{24}(3x-4y)^{6}(3x+28y)^{2}(9x^{2}-8xy+80y^{2})^{6}(225x^{2}-552xy+592y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 132.24.0-6.a.1.8 | $132$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 132.144.1-66.b.1.4 | $132$ | $3$ | $3$ | $1$ |
| 132.96.1-132.i.1.2 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.k.1.2 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.u.1.6 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.w.1.3 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.bg.1.2 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.bi.1.4 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.bo.1.4 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.bq.1.12 | $132$ | $2$ | $2$ | $1$ |
| 264.96.1-264.yv.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zb.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ban.1.14 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bat.1.14 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byi.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byo.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzg.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzm.1.7 | $264$ | $2$ | $2$ | $1$ |