Invariants
| Level: | $312$ | $\SL_2$-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 | $1^{2}\cdot2\cdot4\cdot8^{2}$ | 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
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}4&229\\83&222\end{bmatrix}$, $\begin{bmatrix}49&126\\70&25\end{bmatrix}$, $\begin{bmatrix}201&8\\116&141\end{bmatrix}$, $\begin{bmatrix}210&233\\241&98\end{bmatrix}$, $\begin{bmatrix}229&190\\238&237\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.24.0.cb.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $40255488$ |
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 |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 312.24.0-8.n.1.1 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.0-104.bd.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.be.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bf.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bh.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bj.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bk.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bm.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bp.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dh.2.15 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dj.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dl.2.15 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dn.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ef.2.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ei.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.em.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.er.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.144.4-312.ot.1.43 | $312$ | $3$ | $3$ | $4$ |
| 312.192.3-312.ry.2.37 | $312$ | $4$ | $4$ | $3$ |