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 | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{5}$ | ||
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: | 8N0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}71&22\\148&259\end{bmatrix}$, $\begin{bmatrix}87&62\\164&87\end{bmatrix}$, $\begin{bmatrix}153&28\\176&91\end{bmatrix}$, $\begin{bmatrix}191&86\\164&277\end{bmatrix}$, $\begin{bmatrix}281&66\\296&181\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.48.0.bm.1 for the level structure with $-I$) |
Cyclic 312-isogeny field degree: | $112$ |
Cyclic 312-torsion field degree: | $2688$ |
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 |
---|---|---|---|---|---|
8.48.0-8.e.2.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
312.48.0-8.e.2.12 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.e.1.29 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.e.1.30 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.t.1.29 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.t.1.41 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.bf.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.db.2.10 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.eo.2.10 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ew.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ij.2.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ir.1.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ke.1.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.km.2.3 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.mf.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.mn.2.9 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.oa.2.9 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.oi.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ph.2.1 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.pp.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.qb.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.qf.2.1 | $312$ | $2$ | $2$ | $1$ |
312.288.8-312.ed.1.16 | $312$ | $3$ | $3$ | $8$ |
312.384.7-312.dr.1.34 | $312$ | $4$ | $4$ | $7$ |