Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-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 | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 12I0 |
Level structure
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 |
|---|---|---|---|---|---|
| $X_{\pm1}(2,6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.96.1.le.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lf.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lf.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lg.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lh.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lh.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.li.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.li.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lj.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lj.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lk.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lk.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ll.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ll.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lm.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lm.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ln.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ln.4 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lp.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lp.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lq.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ls.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ls.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lt.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lt.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lv.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lw.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.lw.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.3.ey.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fa.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fb.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fd.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fe.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fg.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fh.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.fj.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ge.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gg.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gh.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gj.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gk.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gm.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gn.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.gp.2 | $312$ | $2$ | $2$ | $3$ |
| 312.144.3.a.1 | $312$ | $3$ | $3$ | $3$ |