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: | $\begin{bmatrix}1&144\\122&143\end{bmatrix}$, $\begin{bmatrix}61&252\\2&149\end{bmatrix}$, $\begin{bmatrix}93&4\\110&47\end{bmatrix}$, $\begin{bmatrix}215&204\\262&55\end{bmatrix}$, $\begin{bmatrix}251&132\\212&137\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 312.48.0.cs.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.m.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ |
104.48.0-104.h.2.29 | $104$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-104.h.2.25 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-24.m.1.18 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.1.33 | $312$ | $2$ | $2$ | $0$ | $?$ |
312.48.0-312.u.1.64 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.n.1.5 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.q.1.11 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.bw.1.10 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.cb.1.5 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.fg.1.10 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.fh.1.7 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.fk.1.12 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.fl.1.9 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.ne.1.6 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nf.1.9 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nq.1.9 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nr.1.7 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nu.1.12 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.nv.1.5 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.og.1.11 | $312$ | $2$ | $2$ | $1$ |
312.192.1-312.oh.1.11 | $312$ | $2$ | $2$ | $1$ |
312.288.8-312.oq.2.25 | $312$ | $3$ | $3$ | $8$ |
312.384.7-312.je.1.55 | $312$ | $4$ | $4$ | $7$ |