Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | ||||
| 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 | $1^{4}\cdot3^{4}\cdot8\cdot24$ | Cusp orbits | $1^{2}\cdot4^{2}$ | ||
| 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: | 24B0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}10&261\\257&176\end{bmatrix}$, $\begin{bmatrix}61&256\\72&161\end{bmatrix}$, $\begin{bmatrix}78&289\\7&36\end{bmatrix}$, $\begin{bmatrix}128&45\\153&110\end{bmatrix}$, $\begin{bmatrix}281&146\\34&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.dp.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| 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 |
|---|---|---|---|---|---|
| 12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-12.f.1.9 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.192.3-312.eo.2.8 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.hu.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.ig.2.15 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.im.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.lm.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.lp.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.lq.2.9 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.lt.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.qo.1.15 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.qr.2.9 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.qs.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.qv.2.9 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.re.2.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.rh.1.15 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.ri.1.13 | $312$ | $2$ | $2$ | $3$ |
| 312.192.3-312.rl.1.15 | $312$ | $2$ | $2$ | $3$ |
| 312.288.3-312.d.1.29 | $312$ | $3$ | $3$ | $3$ |