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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}33&8\\112&59\end{bmatrix}$, $\begin{bmatrix}53&104\\53&115\end{bmatrix}$, $\begin{bmatrix}97&120\\59&135\end{bmatrix}$, $\begin{bmatrix}133&80\\72&65\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 136.24.0.bp.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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.n.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
136.24.0-8.n.1.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.v.1.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.v.1.8 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.y.1.9 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.y.1.16 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
136.96.0-136.bo.1.2 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.bo.2.4 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.bp.1.1 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.bp.2.2 | $136$ | $2$ | $2$ | $0$ |
272.96.0-272.bg.1.2 | $272$ | $2$ | $2$ | $0$ |
272.96.0-272.bg.2.2 | $272$ | $2$ | $2$ | $0$ |
272.96.0-272.bh.1.2 | $272$ | $2$ | $2$ | $0$ |
272.96.0-272.bh.2.2 | $272$ | $2$ | $2$ | $0$ |
272.96.1-272.v.1.2 | $272$ | $2$ | $2$ | $1$ |
272.96.1-272.x.1.2 | $272$ | $2$ | $2$ | $1$ |
272.96.1-272.cl.1.2 | $272$ | $2$ | $2$ | $1$ |
272.96.1-272.cn.1.2 | $272$ | $2$ | $2$ | $1$ |