Invariants
| Level: | $140$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/140\Z)$-generators: | $\begin{bmatrix}27&32\\2&5\end{bmatrix}$, $\begin{bmatrix}105&12\\24&139\end{bmatrix}$, $\begin{bmatrix}131&104\\87&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 140.12.0.g.1 for the level structure with $-I$) |
| Cyclic 140-isogeny field degree: | $48$ |
| Cyclic 140-torsion field degree: | $2304$ |
| Full 140-torsion field degree: | $3870720$ |
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 |
|---|---|---|---|---|---|
| 20.12.0-4.c.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 28.12.0-4.c.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 140.120.4-140.o.1.2 | $140$ | $5$ | $5$ | $4$ |
| 140.144.3-140.s.1.1 | $140$ | $6$ | $6$ | $3$ |
| 140.192.5-140.o.1.1 | $140$ | $8$ | $8$ | $5$ |
| 140.240.7-140.w.1.2 | $140$ | $10$ | $10$ | $7$ |
| 140.504.16-140.w.1.2 | $140$ | $21$ | $21$ | $16$ |
| 280.48.0-280.dc.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dc.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dd.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dd.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.do.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.do.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dp.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dp.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ds.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ds.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dt.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dt.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dw.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dw.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dx.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dx.1.15 | $280$ | $2$ | $2$ | $0$ |