Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $4^{3}\cdot12^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F2 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}131&34\\260&261\end{bmatrix}$, $\begin{bmatrix}135&226\\34&135\end{bmatrix}$, $\begin{bmatrix}141&100\\76&183\end{bmatrix}$, $\begin{bmatrix}201&238\\130&111\end{bmatrix}$, $\begin{bmatrix}209&76\\56&183\end{bmatrix}$, $\begin{bmatrix}279&286\\124&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.2.b.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-6.a.1.11 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.32.0-312.a.1.1 | $312$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.192.3-312.ey.2.24 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.ez.2.28 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fb.1.31 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fc.1.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fe.2.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.ff.2.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fh.1.24 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fi.1.28 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fk.1.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fl.1.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fn.2.24 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fo.2.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fq.1.30 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fr.1.28 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.ft.2.16 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.fu.2.16 | $312$ | $2$ | $2$ | $3$ |
| 312.288.7-312.co.1.30 | $312$ | $3$ | $3$ | $7$ |