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 | $1^{2}\cdot2\cdot4\cdot8^{2}$ | 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: | 8I0 |
Level structure
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}122&135\\161&232\end{bmatrix}$, $\begin{bmatrix}146&89\\89&218\end{bmatrix}$, $\begin{bmatrix}203&116\\326&169\end{bmatrix}$, $\begin{bmatrix}280&307\\91&256\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 328.24.0.ca.1 for the level structure with $-I$) |
| Cyclic 328-isogeny field degree: | $42$ |
| Cyclic 328-torsion field degree: | $6720$ |
| 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 |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 328.24.0-8.n.1.8 | $328$ | $2$ | $2$ | $0$ | $?$ |
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.bb.2.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.be.2.8 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bf.2.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bg.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bi.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bl.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bn.2.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bo.2.4 | $328$ | $2$ | $2$ | $0$ |