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: | $2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}71&104\\76&201\end{bmatrix}$, $\begin{bmatrix}125&160\\182&83\end{bmatrix}$, $\begin{bmatrix}169&68\\166&211\end{bmatrix}$, $\begin{bmatrix}201&280\\166&147\end{bmatrix}$, $\begin{bmatrix}279&212\\310&169\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.48.0.ci.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $5376$ |
Full 312-torsion field degree: | $20127744$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.48.0-24.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
104.48.0-104.l.1.19 | $104$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-24.i.2.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-104.l.1.10 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.2.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.2.30 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
312.192.1-312.bx.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.cf.2.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.cq.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.cx.1.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.do.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.dp.1.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.dq.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.dr.2.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ik.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.il.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.io.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ip.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.jq.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.jr.2.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ju.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.jv.1.4 | $312$ | $2$ | $2$ | $1$ |
312.288.8-312.no.2.34 | $312$ | $3$ | $3$ | $8$ |
312.384.7-312.hs.1.20 | $312$ | $4$ | $4$ | $7$ |