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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}57&104\\122&37\end{bmatrix}$, $\begin{bmatrix}59&88\\118&47\end{bmatrix}$, $\begin{bmatrix}73&120\\12&111\end{bmatrix}$, $\begin{bmatrix}87&112\\40&23\end{bmatrix}$, $\begin{bmatrix}111&8\\28&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.i.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $18$ |
| Cyclic 136-torsion field degree: | $1152$ |
| 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 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 136.24.0-4.b.1.8 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.24.0-8.n.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.j.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.j.2.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.k.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.k.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.l.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.l.2.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bb.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bb.2.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.2.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bd.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bd.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-8.h.1.9 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.p.1.8 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bu.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bv.1.2 | $136$ | $2$ | $2$ | $1$ |
| 272.96.0-16.d.1.11 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-16.d.2.9 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.f.1.18 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.f.2.20 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-16.a.1.10 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-16.a.2.11 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.a.1.19 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.a.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-16.b.1.12 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-16.b.2.9 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.b.1.19 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.b.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.96.2-16.d.1.9 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-16.d.2.12 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.f.1.13 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.f.2.13 | $272$ | $2$ | $2$ | $2$ |