Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot12^{4}\cdot24^{4}$ | Cusp orbits | $1^{4}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AH9 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}29&186\\132&299\end{bmatrix}$, $\begin{bmatrix}169&24\\160&179\end{bmatrix}$, $\begin{bmatrix}215&180\\228&155\end{bmatrix}$, $\begin{bmatrix}215&184\\236&219\end{bmatrix}$, $\begin{bmatrix}239&28\\164&9\end{bmatrix}$, $\begin{bmatrix}279&94\\64&225\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.192.9.jl.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 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 104.96.1-104.bf.2.4 | $104$ | $4$ | $4$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.3-24.bq.2.47 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 104.96.1-104.bf.2.4 | $104$ | $4$ | $4$ | $1$ | $?$ |
| 312.192.3-24.bq.2.2 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.ew.1.8 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.ew.1.55 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.5-312.c.1.22 | $312$ | $2$ | $2$ | $5$ | $?$ |
| 312.192.5-312.c.1.34 | $312$ | $2$ | $2$ | $5$ | $?$ |