Invariants
Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-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 | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | 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: | 24Z5 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}7&240\\185&91\end{bmatrix}$, $\begin{bmatrix}205&168\\30&103\end{bmatrix}$, $\begin{bmatrix}229&132\\239&71\end{bmatrix}$, $\begin{bmatrix}301&36\\150&91\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.192.5.bay.2 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $28$ |
Cyclic 312-torsion field degree: | $2688$ |
Full 312-torsion field degree: | $5031936$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.192.1-24.dm.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ |
156.192.1-156.m.4.4 | $156$ | $2$ | $2$ | $1$ | $?$ |
312.192.1-156.m.4.12 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.192.1-24.dm.2.16 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.192.1-312.rk.4.1 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.192.1-312.rk.4.16 | $312$ | $2$ | $2$ | $1$ | $?$ |
312.192.3-312.ne.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.ne.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.pb.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.pb.1.28 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.st.3.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.st.3.16 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.tb.4.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
312.192.3-312.tb.4.24 | $312$ | $2$ | $2$ | $3$ | $?$ |