Invariants
| Level: | $304$ | $\SL_2$-level: | $16$ | ||||
| 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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/304\Z)$-generators: | $\begin{bmatrix}123&32\\220&7\end{bmatrix}$, $\begin{bmatrix}133&108\\112&129\end{bmatrix}$, $\begin{bmatrix}144&123\\37&78\end{bmatrix}$, $\begin{bmatrix}278&293\\241&162\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 304.48.0.bc.1 for the level structure with $-I$) |
| Cyclic 304-isogeny field degree: | $40$ |
| Cyclic 304-torsion field degree: | $1440$ |
| Full 304-torsion field degree: | $31518720$ |
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 |
|---|---|---|---|---|---|
| 16.48.0-16.e.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 152.48.0-152.bu.2.4 | $152$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-16.e.1.14 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-304.g.1.6 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-304.g.1.32 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-152.bu.2.2 | $304$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 304.192.1-304.p.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.u.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bg.2.5 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bv.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.cj.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.cm.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.cy.2.3 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.dd.2.5 | $304$ | $2$ | $2$ | $1$ |