Invariants
| Level: | $312$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{6}$ | 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: | 4G0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}17&276\\16&37\end{bmatrix}$, $\begin{bmatrix}119&274\\80&69\end{bmatrix}$, $\begin{bmatrix}139&68\\24&133\end{bmatrix}$, $\begin{bmatrix}141&292\\176&189\end{bmatrix}$, $\begin{bmatrix}161&210\\188&253\end{bmatrix}$, $\begin{bmatrix}303&256\\176&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.24.0.e.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| 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 |
|---|---|---|---|---|---|
| 4.24.0-4.b.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 312.24.0-312.a.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-312.a.1.9 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-4.b.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-312.b.1.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-312.b.1.9 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.