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 $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8J0 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}23&112\\100&95\end{bmatrix}$, $\begin{bmatrix}47&128\\80&69\end{bmatrix}$, $\begin{bmatrix}77&60\\76&67\end{bmatrix}$, $\begin{bmatrix}97&24\\10&17\end{bmatrix}$, $\begin{bmatrix}103&4\\90&5\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.24.0.d.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 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
136.24.0-4.b.1.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-4.b.1.4 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.a.1.3 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.b.2.4 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.d.1.7 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.e.1.8 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.g.1.4 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.h.1.2 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.i.2.14 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.j.1.1 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.j.2.7 | $136$ | $2$ | $2$ | $0$ |
136.96.0-8.k.2.3 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.m.2.16 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.n.2.14 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.q.1.7 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.r.2.3 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.u.2.1 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.v.1.6 | $136$ | $2$ | $2$ | $0$ |
136.96.1-8.e.2.5 | $136$ | $2$ | $2$ | $1$ |
136.96.1-8.i.1.3 | $136$ | $2$ | $2$ | $1$ |
136.96.1-8.l.1.7 | $136$ | $2$ | $2$ | $1$ |
136.96.1-8.m.2.8 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bc.2.6 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bd.2.15 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bg.2.6 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bh.2.8 | $136$ | $2$ | $2$ | $1$ |