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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}31&8\\124&81\end{bmatrix}$, $\begin{bmatrix}41&44\\134&43\end{bmatrix}$, $\begin{bmatrix}105&120\\130&113\end{bmatrix}$, $\begin{bmatrix}113&4\\90&119\end{bmatrix}$, $\begin{bmatrix}127&16\\0&59\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.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 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 136.24.0-4.b.1.6 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.24.0-4.b.1.8 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.96.0-8.b.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.c.1.9 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.e.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.f.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.h.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.i.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.k.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.k.2.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.l.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.l.2.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.o.2.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.p.2.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.s.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.t.1.16 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.w.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.x.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-8.i.2.8 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.k.2.6 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.m.2.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.n.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.be.2.8 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bf.2.11 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bi.2.5 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bj.2.4 | $136$ | $2$ | $2$ | $1$ |