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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}15&80\\182&215\end{bmatrix}$, $\begin{bmatrix}33&212\\272&155\end{bmatrix}$, $\begin{bmatrix}87&20\\146&133\end{bmatrix}$, $\begin{bmatrix}193&136\\166&51\end{bmatrix}$, $\begin{bmatrix}311&268\\290&153\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.be.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $20127744$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-8.e.1.2 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-156.c.1.9 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-156.c.1.23 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.1.32 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.1.58 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.be.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.da.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dy.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eg.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.iz.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jh.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jo.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jw.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mu.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nc.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nl.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nt.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pg.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.po.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pt.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.px.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.dn.2.7 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.dj.2.18 | $312$ | $4$ | $4$ | $7$ |