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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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
| $\GL_2(\Z/304\Z)$-generators: | $\begin{bmatrix}21&128\\89&51\end{bmatrix}$, $\begin{bmatrix}35&280\\152&151\end{bmatrix}$, $\begin{bmatrix}137&16\\50&301\end{bmatrix}$, $\begin{bmatrix}149&64\\292&69\end{bmatrix}$, $\begin{bmatrix}183&248\\127&249\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 152.48.0.bd.1 for the level structure with $-I$) |
| Cyclic 304-isogeny field degree: | $40$ |
| Cyclic 304-torsion field degree: | $2880$ |
| 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 |
|---|---|---|---|---|---|
| 16.48.0-8.q.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 304.48.0-8.q.1.1 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-152.bu.1.5 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-152.bu.1.16 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-152.bv.2.3 | $304$ | $2$ | $2$ | $0$ | $?$ |
| 304.48.0-152.bv.2.16 | $304$ | $2$ | $2$ | $0$ | $?$ |
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.bf.2.3 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bg.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bh.2.5 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bk.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bl.1.3 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bo.2.5 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bp.1.3 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-304.bq.2.2 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-152.cj.2.3 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-152.ck.1.7 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-152.cl.1.6 | $304$ | $2$ | $2$ | $1$ |
| 304.192.1-152.cm.2.7 | $304$ | $2$ | $2$ | $1$ |
| 304.192.3-304.fq.1.7 | $304$ | $2$ | $2$ | $3$ |
| 304.192.3-304.fs.2.7 | $304$ | $2$ | $2$ | $3$ |
| 304.192.3-304.fy.1.6 | $304$ | $2$ | $2$ | $3$ |
| 304.192.3-304.gc.2.7 | $304$ | $2$ | $2$ | $3$ |