Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}7&64\\100&81\end{bmatrix}$, $\begin{bmatrix}23&52\\12&37\end{bmatrix}$, $\begin{bmatrix}79&48\\120&61\end{bmatrix}$, $\begin{bmatrix}97&4\\16&67\end{bmatrix}$, $\begin{bmatrix}111&104\\12&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.c.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $1152$ |
| Full 136-torsion field degree: | $1253376$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{48}(x^{8}-4x^{7}y+4x^{6}y^{2}+28x^{5}y^{3}+6x^{4}y^{4}-28x^{3}y^{5}+4x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+4x^{7}y+4x^{6}y^{2}-28x^{5}y^{3}+6x^{4}y^{4}+28x^{3}y^{5}+4x^{2}y^{6}-4xy^{7}+y^{8})^{3}}{y^{4}x^{52}(x-y)^{4}(x+y)^{4}(x^{2}+y^{2})^{8}(x^{2}-2xy-y^{2})^{4}(x^{2}+2xy-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 136.48.0-4.b.1.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-4.b.1.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.1.3 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.1.6 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.2.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.2.14 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.192.1-8.f.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.f.2.1 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.g.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.g.2.1 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.w.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.w.2.7 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.x.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.x.2.7 | $136$ | $2$ | $2$ | $1$ |
| 136.192.3-8.i.1.3 | $136$ | $2$ | $2$ | $3$ |
| 136.192.3-8.j.1.3 | $136$ | $2$ | $2$ | $3$ |
| 136.192.3-136.be.1.9 | $136$ | $2$ | $2$ | $3$ |
| 136.192.3-136.bf.1.9 | $136$ | $2$ | $2$ | $3$ |
| 272.192.2-16.a.1.10 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.a.1.9 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-16.b.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.b.1.10 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-16.c.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-16.d.1.5 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.k.1.2 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.l.1.7 | $272$ | $2$ | $2$ | $2$ |