Invariants
| Level: | $140$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-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: | $\begin{bmatrix}29&58\\70&51\end{bmatrix}$, $\begin{bmatrix}31&122\\10&23\end{bmatrix}$, $\begin{bmatrix}39&138\\12&33\end{bmatrix}$, $\begin{bmatrix}49&20\\52&119\end{bmatrix}$, $\begin{bmatrix}99&44\\34&41\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 140.24.0-140.a.1.1, 140.24.0-140.a.1.2, 140.24.0-140.a.1.3, 140.24.0-140.a.1.4, 140.24.0-140.a.1.5, 140.24.0-140.a.1.6, 140.24.0-140.a.1.7, 140.24.0-140.a.1.8, 280.24.0-140.a.1.1, 280.24.0-140.a.1.2, 280.24.0-140.a.1.3, 280.24.0-140.a.1.4, 280.24.0-140.a.1.5, 280.24.0-140.a.1.6, 280.24.0-140.a.1.7, 280.24.0-140.a.1.8 |
| Cyclic 140-isogeny field degree: | $96$ |
| Cyclic 140-torsion field degree: | $4608$ |
| Full 140-torsion field degree: | $7741440$ |
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 |
|---|---|---|---|---|---|
| $X(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
| 140.6.0.b.1 | $140$ | $2$ | $2$ | $0$ | $?$ |
| 140.6.0.e.1 | $140$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 140.24.0.a.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.c.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.d.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.g.1 | $140$ | $2$ | $2$ | $0$ |
| 140.60.4.c.1 | $140$ | $5$ | $5$ | $4$ |
| 140.72.3.c.1 | $140$ | $6$ | $6$ | $3$ |
| 140.96.5.c.1 | $140$ | $8$ | $8$ | $5$ |
| 140.120.7.c.1 | $140$ | $10$ | $10$ | $7$ |
| 140.252.16.c.1 | $140$ | $21$ | $21$ | $16$ |
| 140.336.21.c.1 | $140$ | $28$ | $28$ | $21$ |
| 280.24.0.b.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.f.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.j.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.s.1 | $280$ | $2$ | $2$ | $0$ |