Invariants
| Level: | $328$ | $\SL_2$-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
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}89&32\\152&173\end{bmatrix}$, $\begin{bmatrix}129&100\\14&309\end{bmatrix}$, $\begin{bmatrix}265&280\\182&295\end{bmatrix}$, $\begin{bmatrix}287&144\\302&283\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 328.48.0.k.2 for the level structure with $-I$) |
| Cyclic 328-isogeny field degree: | $84$ |
| Cyclic 328-torsion field degree: | $13440$ |
| Full 328-torsion field degree: | $44083200$ |
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.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 328.48.0-8.e.1.2 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-164.c.1.4 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-164.c.1.13 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.h.1.16 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.h.1.30 | $328$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 328.192.1-328.i.1.8 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.y.1.2 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.bc.1.4 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.bg.1.6 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.bu.1.6 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.by.1.4 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.cb.1.2 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.cd.1.8 | $328$ | $2$ | $2$ | $1$ |