Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| 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: | 8C0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}13&102\\116&23\end{bmatrix}$, $\begin{bmatrix}22&115\\99&54\end{bmatrix}$, $\begin{bmatrix}60&121\\7&6\end{bmatrix}$, $\begin{bmatrix}62&17\\85&58\end{bmatrix}$, $\begin{bmatrix}97&132\\8&77\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $18$ |
| Cyclic 136-torsion field degree: | $1152$ |
| Full 136-torsion field degree: | $5013504$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 68.12.0-4.c.1.2 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 136.12.0-4.c.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 |
|---|---|---|---|---|
| 136.48.0-8.i.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.k.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.q.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.r.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.ba.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.ba.2.5 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.bb.1.7 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-8.bb.2.6 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bj.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bl.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bn.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bp.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.ca.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.ca.2.16 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.cb.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.cb.2.16 | $136$ | $2$ | $2$ | $0$ |
| 136.432.15-136.cj.1.33 | $136$ | $18$ | $18$ | $15$ |
| 272.48.0-16.e.1.3 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-16.e.2.6 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-16.f.1.4 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-16.f.2.5 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-16.g.1.5 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-16.h.1.5 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.m.1.11 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.m.2.23 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.n.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.n.2.27 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.o.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.p.1.3 | $272$ | $2$ | $2$ | $0$ |
| 272.48.1-16.a.1.12 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-272.a.1.30 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-16.b.1.12 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-272.b.1.26 | $272$ | $2$ | $2$ | $1$ |