Invariants
Level: | $312$ | $\SL_2$-level: | $4$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4G0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}7&270\\202&119\end{bmatrix}$, $\begin{bmatrix}161&8\\54&95\end{bmatrix}$, $\begin{bmatrix}211&146\\288&191\end{bmatrix}$, $\begin{bmatrix}215&226\\278&155\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.24.0.q.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $224$ |
Cyclic 312-torsion field degree: | $10752$ |
Full 312-torsion field degree: | $40255488$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.24.0-12.b.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
104.24.0-104.b.1.5 | $104$ | $2$ | $2$ | $0$ | $?$ |
312.24.0-312.a.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.24.0-312.a.1.13 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.24.0-12.b.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.24.0-104.b.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.144.4-312.x.1.30 | $312$ | $3$ | $3$ | $4$ |
312.192.3-312.el.1.3 | $312$ | $4$ | $4$ | $3$ |