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 $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
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: | 8N0 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}7&58\\116&75\end{bmatrix}$, $\begin{bmatrix}65&36\\20&7\end{bmatrix}$, $\begin{bmatrix}97&62\\60&131\end{bmatrix}$, $\begin{bmatrix}123&70\\96&11\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 136.48.0.o.2 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 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 |
---|---|---|---|---|---|
8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
136.48.0-8.e.1.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.48.0-136.e.1.17 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.48.0-136.e.1.20 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.48.0-136.h.2.14 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.48.0-136.h.2.26 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
136.192.1-136.j.2.7 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.z.1.5 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.bk.1.5 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.bo.2.7 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.bv.1.6 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.bz.2.8 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.cf.2.8 | $136$ | $2$ | $2$ | $1$ |
136.192.1-136.ch.1.6 | $136$ | $2$ | $2$ | $1$ |