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 | $2^{3}\cdot4$ | ||
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: | 8N0 |
Level structure
$\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}23&224\\284&97\end{bmatrix}$, $\begin{bmatrix}91&16\\210&193\end{bmatrix}$, $\begin{bmatrix}241&120\\288&227\end{bmatrix}$, $\begin{bmatrix}259&116\\142&187\end{bmatrix}$, $\begin{bmatrix}281&192\\142&19\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.48.0.u.2 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 | $24$ | $2$ | $2$ | $0$ | $0$ |
104.48.0-52.c.1.2 | $104$ | $2$ | $2$ | $0$ | $?$ |
156.48.0-52.c.1.1 | $156$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-24.i.2.23 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.1.8 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.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.cc.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.cu.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.do.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.dq.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.fz.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.gd.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.he.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.hi.1.4 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.mg.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.mk.1.7 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nm.1.7 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nq.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.oo.1.7 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.oq.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ou.1.8 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ov.1.7 | $312$ | $2$ | $2$ | $1$ |
312.288.8-312.ck.2.18 | $312$ | $3$ | $3$ | $8$ |
312.384.7-312.cl.1.62 | $312$ | $4$ | $4$ | $7$ |