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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
Level structure
| $\GL_2(\Z/328\Z)$-generators: | $\begin{bmatrix}25&8\\324&243\end{bmatrix}$, $\begin{bmatrix}73&164\\180&115\end{bmatrix}$, $\begin{bmatrix}111&68\\84&93\end{bmatrix}$, $\begin{bmatrix}311&188\\42&261\end{bmatrix}$, $\begin{bmatrix}319&288\\252&201\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.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, including 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 164.24.0-4.b.1.2 | $164$ | $2$ | $2$ | $0$ | $?$ |
| 328.24.0-4.b.1.2 | $328$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 328.96.0-8.b.1.11 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.c.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.e.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.f.1.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.h.1.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.i.1.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.k.1.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.k.2.13 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-8.l.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.l.2.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.o.2.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.p.2.9 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.s.1.15 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.t.1.15 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.1.10 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.1.16 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-8.i.2.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-8.k.2.5 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-8.m.2.8 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-8.n.1.4 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.be.2.11 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bf.2.7 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bi.2.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bj.2.3 | $328$ | $2$ | $2$ | $1$ |