Invariants
| Level: | $328$ | $\SL_2$-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
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}53&28\\86&197\end{bmatrix}$, $\begin{bmatrix}81&92\\288&43\end{bmatrix}$, $\begin{bmatrix}225&88\\210&301\end{bmatrix}$, $\begin{bmatrix}231&180\\230&235\end{bmatrix}$, $\begin{bmatrix}269&244\\318&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 328.24.0.m.1 for the level structure with $-I$) |
| Cyclic 328-isogeny field degree: | $84$ |
| Cyclic 328-torsion field degree: | $13440$ |
| Full 328-torsion field degree: | $88166400$ |
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 |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 328.24.0-4.b.1.6 | $328$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 328.96.0-328.u.1.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.u.1.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.u.2.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.u.2.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.1.10 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.1.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.2.10 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.2.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.1.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.1.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.2.10 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.2.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.1.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.1.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.2.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.2.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.o.2.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.o.2.10 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.p.1.5 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.p.1.10 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.ba.1.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.ba.1.10 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bb.1.5 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bb.1.10 | $328$ | $2$ | $2$ | $1$ |