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: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}103&186\\0&49\end{bmatrix}$, $\begin{bmatrix}104&9\\155&262\end{bmatrix}$, $\begin{bmatrix}146&201\\213&2\end{bmatrix}$, $\begin{bmatrix}153&40\\122&31\end{bmatrix}$, $\begin{bmatrix}211&24\\310&269\end{bmatrix}$, $\begin{bmatrix}243&212\\202&89\end{bmatrix}$, $\begin{bmatrix}309&284\\178&311\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $28$ |
| 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, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 312.24.0-6.a.1.3 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-6.a.1.15 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.