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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Level structure
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}117&172\\14&287\end{bmatrix}$, $\begin{bmatrix}183&184\\14&141\end{bmatrix}$, $\begin{bmatrix}261&204\\44&115\end{bmatrix}$, $\begin{bmatrix}283&208\\78&197\end{bmatrix}$, $\begin{bmatrix}323&180\\4&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 328.24.0.i.2 for the level structure with $-I$) |
| Cyclic 328-isogeny field degree: | $84$ |
| 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 |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 328.24.0-4.b.1.7 | $328$ | $2$ | $2$ | $0$ | $?$ |
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.b.2.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.c.1.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.e.1.15 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.f.1.9 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.j.1.12 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.l.2.9 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.n.2.9 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.p.1.14 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.r.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.t.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.z.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.ba.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bc.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bd.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.q.2.9 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.s.1.7 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.x.1.15 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.y.1.6 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bd.1.11 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bf.2.9 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bh.2.11 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bj.1.12 | $328$ | $2$ | $2$ | $1$ |