Invariants
Level: | $132$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 4E0 |
Level structure
$\GL_2(\Z/132\Z)$-generators: | $\begin{bmatrix}31&88\\73&101\end{bmatrix}$, $\begin{bmatrix}105&128\\1&55\end{bmatrix}$, $\begin{bmatrix}117&52\\130&71\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.12.0.h.1 for the level structure with $-I$) |
Cyclic 132-isogeny field degree: | $48$ |
Cyclic 132-torsion field degree: | $1920$ |
Full 132-torsion field degree: | $2534400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 771 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3}\cdot\frac{(3x-2y)^{12}(9x^{4}+42x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(3x-2y)^{12}(3x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
44.12.0-4.c.1.2 | $44$ | $2$ | $2$ | $0$ | $0$ |
132.12.0-4.c.1.1 | $132$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
132.72.2-12.t.1.1 | $132$ | $3$ | $3$ | $2$ |
132.96.1-12.l.1.1 | $132$ | $4$ | $4$ | $1$ |
132.288.9-132.l.1.2 | $132$ | $12$ | $12$ | $9$ |
264.48.0-24.bk.1.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bk.1.6 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bl.1.6 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bl.1.9 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bs.1.4 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bs.1.5 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bt.1.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bt.1.5 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.by.1.4 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.by.1.13 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.bz.1.4 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.bz.1.9 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cc.1.4 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cc.1.13 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cd.1.8 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cd.1.10 | $264$ | $2$ | $2$ | $0$ |