Invariants
Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12L3 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}3&256\\136&261\end{bmatrix}$, $\begin{bmatrix}29&304\\242&3\end{bmatrix}$, $\begin{bmatrix}143&94\\20&93\end{bmatrix}$, $\begin{bmatrix}153&4\\268&303\end{bmatrix}$, $\begin{bmatrix}203&166\\242&105\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.96.3.fi.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $5376$ |
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 |
---|---|---|---|---|---|
24.96.1-24.bz.1.20 | $24$ | $2$ | $2$ | $1$ | $1$ |
156.96.0-156.a.1.8 | $156$ | $2$ | $2$ | $0$ | $?$ |
312.96.0-156.a.1.20 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.96.1-24.bz.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.96.2-312.b.1.9 | $312$ | $2$ | $2$ | $2$ | $?$ |
312.96.2-312.b.1.24 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.384.5-312.ic.2.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.id.2.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.id.3.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ih.2.11 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ih.3.12 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ii.2.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ii.3.6 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kl.2.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kl.4.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.km.2.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.km.4.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ks.3.4 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.ks.4.7 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kt.2.8 | $312$ | $2$ | $2$ | $5$ |
312.384.5-312.kt.4.8 | $312$ | $2$ | $2$ | $5$ |