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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}45&100\\16&59\end{bmatrix}$, $\begin{bmatrix}47&0\\124&53\end{bmatrix}$, $\begin{bmatrix}73&40\\110&105\end{bmatrix}$, $\begin{bmatrix}105&64\\120&131\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.h.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $2304$ |
| 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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(x-y)^{48}(x^{8}-4x^{7}y+16x^{6}y^{2}-56x^{5}y^{3}+120x^{4}y^{4}-112x^{3}y^{5}+64x^{2}y^{6}-32xy^{7}+16y^{8})^{3}(13x^{8}-140x^{7}y+688x^{6}y^{2}-1960x^{5}y^{3}+3480x^{4}y^{4}-3920x^{3}y^{5}+2752x^{2}y^{6}-1120xy^{7}+208y^{8})^{3}}{(x-y)^{48}(x^{2}-2y^{2})^{4}(x^{2}-4xy+2y^{2})^{8}(x^{2}-4xy+6y^{2})^{2}(x^{2}-2xy+2y^{2})^{8}(3x^{2}-4xy+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 136.48.0-8.d.1.9 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.d.1.11 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.1.10 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.1.11 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.h.1.8 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.h.1.10 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.a.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.c.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.f.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-8.h.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bu.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bv.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bw.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bx.1.2 | $136$ | $2$ | $2$ | $1$ |