Invariants
Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $12^{12}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $4 \le \gamma \le 12$ | ||||||
$\overline{\Q}$-gonality: | $4 \le \gamma \le 7$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12B7 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}23&204\\36&233\end{bmatrix}$, $\begin{bmatrix}29&262\\172&219\end{bmatrix}$, $\begin{bmatrix}121&136\\6&11\end{bmatrix}$, $\begin{bmatrix}201&122\\148&201\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.144.7.pn.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $10752$ |
Full 312-torsion field degree: | $6709248$ |
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 |
---|---|---|---|---|---|
24.144.4-24.bb.1.5 | $24$ | $2$ | $2$ | $4$ | $1$ |
156.144.3-156.bh.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ |
312.144.3-312.z.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.144.3-312.z.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.144.3-156.bh.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.144.3-312.dv.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.144.3-312.dv.1.12 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.144.4-312.k.1.9 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-312.k.1.10 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-24.bb.1.7 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-312.di.1.29 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-312.di.1.30 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-312.dk.1.11 | $312$ | $2$ | $2$ | $4$ | $?$ |
312.144.4-312.dk.1.12 | $312$ | $2$ | $2$ | $4$ | $?$ |