Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | ||||
| 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 | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 12I0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}53&90\\194&199\end{bmatrix}$, $\begin{bmatrix}69&20\\64&83\end{bmatrix}$, $\begin{bmatrix}69&310\\112&63\end{bmatrix}$, $\begin{bmatrix}167&304\\302&15\end{bmatrix}$, $\begin{bmatrix}181&42\\54&283\end{bmatrix}$, $\begin{bmatrix}293&202\\68&135\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.o.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
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 |
|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-6.a.1.2 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.