Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | ||||
| 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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}68&115\\231&208\end{bmatrix}$, $\begin{bmatrix}117&218\\236&135\end{bmatrix}$, $\begin{bmatrix}138&79\\259&156\end{bmatrix}$, $\begin{bmatrix}224&303\\267&218\end{bmatrix}$, $\begin{bmatrix}258&199\\167&232\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.24.0.ft.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $2688$ |
| 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 |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 312.24.0-6.a.1.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.1-312.dj.1.36 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.gj.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.kc.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ke.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bkw.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bky.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.blf.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.blh.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byk.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bym.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byt.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byv.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzu.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzw.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.cad.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.caf.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.144.1-312.ca.1.8 | $312$ | $3$ | $3$ | $1$ |