Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}43&120\\232&299\end{bmatrix}$, $\begin{bmatrix}133&167\\169&60\end{bmatrix}$, $\begin{bmatrix}199&104\\262&75\end{bmatrix}$, $\begin{bmatrix}212&55\\75&217\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.16.0.a.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $168$ |
| Cyclic 312-torsion field degree: | $16128$ |
| Full 312-torsion field degree: | $60383232$ |
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 |
|---|---|---|---|---|---|
| 24.16.0-6.a.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 156.16.0-6.a.1.4 | $156$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.2-312.b.1.4 | $312$ | $3$ | $3$ | $2$ |
| 312.96.2-312.j.2.7 | $312$ | $3$ | $3$ | $2$ |
| 312.96.2-312.k.1.3 | $312$ | $3$ | $3$ | $2$ |
| 312.96.3-312.a.1.9 | $312$ | $3$ | $3$ | $3$ |
| 312.128.1-312.a.1.2 | $312$ | $4$ | $4$ | $1$ |
| 312.448.15-312.c.2.5 | $312$ | $14$ | $14$ | $15$ |