Invariants
Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E5 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}83&22\\0&205\end{bmatrix}$, $\begin{bmatrix}147&136\\308&115\end{bmatrix}$, $\begin{bmatrix}167&118\\122&285\end{bmatrix}$, $\begin{bmatrix}225&190\\172&249\end{bmatrix}$, $\begin{bmatrix}251&10\\234&301\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 312.384.5-312.kt.4.1, 312.384.5-312.kt.4.2, 312.384.5-312.kt.4.3, 312.384.5-312.kt.4.4, 312.384.5-312.kt.4.5, 312.384.5-312.kt.4.6, 312.384.5-312.kt.4.7, 312.384.5-312.kt.4.8, 312.384.5-312.kt.4.9, 312.384.5-312.kt.4.10, 312.384.5-312.kt.4.11, 312.384.5-312.kt.4.12, 312.384.5-312.kt.4.13, 312.384.5-312.kt.4.14, 312.384.5-312.kt.4.15, 312.384.5-312.kt.4.16 |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $2688$ |
Full 312-torsion field degree: | $10063872$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.1.cp.2 | $24$ | $2$ | $2$ | $1$ | $1$ |
156.96.1.g.3 | $156$ | $2$ | $2$ | $1$ | $?$ |
312.96.1.lr.2 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.3.en.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.96.3.fi.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.96.3.fn.2 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.96.3.fr.1 | $312$ | $2$ | $2$ | $3$ | $?$ |