Invariants
| Level: | $312$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}29&158\\42&65\end{bmatrix}$, $\begin{bmatrix}31&302\\160&301\end{bmatrix}$, $\begin{bmatrix}115&214\\236&73\end{bmatrix}$, $\begin{bmatrix}116&233\\207&214\end{bmatrix}$, $\begin{bmatrix}266&271\\121&40\end{bmatrix}$, $\begin{bmatrix}271&76\\98&273\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $161021952$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.24.0-4.b.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-4.d.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.d.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-12.g.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-52.g.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-156.g.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-12.h.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-52.h.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-156.h.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.k.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.m.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.m.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.n.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.n.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.o.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.o.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.p.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-8.p.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.s.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.s.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.s.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.v.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.v.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.v.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.y.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.y.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.y.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.y.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.y.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.y.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.z.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.z.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.z.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.z.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.z.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.z.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.ba.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.ba.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.ba.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.ba.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.ba.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.ba.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.bb.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-24.bb.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.bb.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-104.bb.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.bb.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.bb.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.36.1-12.c.1.5 | $312$ | $3$ | $3$ | $1$ |
| 312.48.0-12.g.1.7 | $312$ | $4$ | $4$ | $0$ |
| 312.168.5-52.c.1.5 | $312$ | $14$ | $14$ | $5$ |