Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| 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 | $4^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 4G0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}2&75\\119&42\end{bmatrix}$, $\begin{bmatrix}12&33\\1&32\end{bmatrix}$, $\begin{bmatrix}35&116\\96&95\end{bmatrix}$, $\begin{bmatrix}135&52\\24&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 68.24.0.d.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $2304$ |
| Full 136-torsion field degree: | $2506752$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{10}\cdot17^2}\cdot\frac{x^{24}(289x^{4}-8704x^{2}y^{2}+16384y^{4})^{3}(289x^{4}+8704x^{2}y^{2}+16384y^{4})^{3}}{y^{4}x^{28}(289x^{4}+16384y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.24.0-4.d.1.5 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.96.1-136.cv.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.cx.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.di.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.dl.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.dx.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.ec.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.em.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.eo.1.1 | $136$ | $2$ | $2$ | $1$ |