Invariants
Level: | $312$ | $\SL_2$-level: | $8$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8O0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}237&280\\13&113\end{bmatrix}$, $\begin{bmatrix}245&192\\203&283\end{bmatrix}$, $\begin{bmatrix}269&272\\24&287\end{bmatrix}$, $\begin{bmatrix}289&176\\307&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.48.0.dg.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $56$ |
Cyclic 312-torsion field degree: | $5376$ |
Full 312-torsion field degree: | $20127744$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
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 |
---|---|---|---|---|---|
24.48.0-24.bh.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
104.48.0-104.ca.1.6 | $104$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-24.bh.1.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-104.ca.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.ei.1.19 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.ei.1.20 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.288.8-312.qk.2.2 | $312$ | $3$ | $3$ | $8$ |
312.384.7-312.kr.1.10 | $312$ | $4$ | $4$ | $7$ |