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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}133&0\\122&281\end{bmatrix}$, $\begin{bmatrix}213&56\\76&25\end{bmatrix}$, $\begin{bmatrix}301&312\\242&85\end{bmatrix}$, $\begin{bmatrix}321&188\\316&311\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 328.48.0.bc.1 for the level structure with $-I$) |
| Cyclic 328-isogeny field degree: | $42$ |
| Cyclic 328-torsion field degree: | $6720$ |
| 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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 328.48.0-8.i.1.8 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.h.1.1 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.h.1.3 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.i.2.7 | $328$ | $2$ | $2$ | $0$ | $?$ |
| 328.48.0-328.i.2.15 | $328$ | $2$ | $2$ | $0$ | $?$ |
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.f.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.q.2.7 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.bq.2.7 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.br.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.ca.2.7 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.cd.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.ce.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.192.1-328.ch.2.7 | $328$ | $2$ | $2$ | $1$ |