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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}71&46\\234&73\end{bmatrix}$, $\begin{bmatrix}75&100\\106&177\end{bmatrix}$, $\begin{bmatrix}159&284\\128&177\end{bmatrix}$, $\begin{bmatrix}163&282\\166&263\end{bmatrix}$, $\begin{bmatrix}165&266\\46&281\end{bmatrix}$, $\begin{bmatrix}187&270\\40&77\end{bmatrix}$, $\begin{bmatrix}199&2\\280&255\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.a.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| 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 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 312.12.0-2.a.1.1 | $312$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.0-12.a.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-12.a.2.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.18 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.1.30 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.19 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-156.a.2.22 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.20 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.24 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.27 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.1.31 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.19 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.o.2.27 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.22 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.32 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.33 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.34 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.1.44 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.28 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.33 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.34 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.56 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-12.a.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.a.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.a.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.a.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.a.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.b.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.b.1.13 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.b.1.17 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.b.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.b.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.b.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.c.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.c.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.c.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.c.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.c.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.c.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.d.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.d.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-12.d.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.d.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.d.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-156.d.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bw.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bw.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bw.1.13 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bx.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bx.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bx.1.13 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.by.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.by.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.by.1.18 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bz.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bz.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.bz.1.18 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dg.1.17 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dg.1.20 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dg.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dh.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dh.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dh.1.31 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.di.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.di.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.di.1.32 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dj.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dj.1.17 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dj.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-12.a.1.3 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-12.a.1.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-12.a.2.2 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-12.a.2.3 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-156.a.1.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-156.a.1.6 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-156.a.2.7 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-156.a.2.8 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.b.1.2 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.b.1.6 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.b.2.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.b.2.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.b.1.17 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.b.1.18 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.b.2.13 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.b.2.14 | $312$ | $2$ | $2$ | $2$ |
| 312.144.1-6.a.1.3 | $312$ | $3$ | $3$ | $1$ |