Invariants
| Level: | $304$ | $\SL_2$-level: | $16$ | ||||
| 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
| $\GL_2(\Z/304\Z)$-generators: | $\begin{bmatrix}46&133\\157&142\end{bmatrix}$, $\begin{bmatrix}137&144\\268&285\end{bmatrix}$, $\begin{bmatrix}140&37\\7&42\end{bmatrix}$, $\begin{bmatrix}199&22\\286&15\end{bmatrix}$, $\begin{bmatrix}281&90\\62&269\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 152.24.0.bv.1 for the level structure with $-I$) |
| Cyclic 304-isogeny field degree: | $40$ |
| Cyclic 304-torsion field degree: | $2880$ |
| Full 304-torsion field degree: | $63037440$ |
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 |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 304.24.0-8.n.1.5 | $304$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 304.96.0-152.bb.1.4 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bc.2.3 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bd.2.6 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bf.1.3 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bh.2.6 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bi.2.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bk.2.8 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bn.2.6 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bd.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bj.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bl.2.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.br.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bt.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bv.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bx.2.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bz.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.bh.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bj.2.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bl.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bn.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bp.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bv.2.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bx.1.1 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.cd.1.1 | $304$ | $2$ | $2$ | $1$ |