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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}101&4\\260&11\end{bmatrix}$, $\begin{bmatrix}161&100\\228&59\end{bmatrix}$, $\begin{bmatrix}161&244\\156&205\end{bmatrix}$, $\begin{bmatrix}161&300\\20&283\end{bmatrix}$, $\begin{bmatrix}259&188\\134&251\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.bu.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
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.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-8.e.2.12 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.1.40 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.1.41 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.x.1.25 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.x.1.34 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.ba.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.bf.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cy.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.db.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dy.2.12 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dz.2.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eg.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eh.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gm.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gn.2.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gu.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gv.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hc.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hd.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hk.2.11 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hl.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.mf.1.40 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.gt.1.51 | $312$ | $4$ | $4$ | $7$ |