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: | $\begin{bmatrix}19&4\\85&121\end{bmatrix}$, $\begin{bmatrix}65&12\\97&125\end{bmatrix}$, $\begin{bmatrix}131&16\\28&115\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.12.0.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, including 631 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7}\cdot\frac{(x+8y)^{12}(3x^{4}-784x^{2}y^{2}-6272xy^{3}-12544y^{4})^{3}}{x^{2}(x+8y)^{14}(x^{2}+14xy+56y^{2})^{4}}$ |
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 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 140.12.0-4.c.1.2 | $140$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 140.120.4-140.k.1.4 | $140$ | $5$ | $5$ | $4$ |
| 140.144.3-140.o.1.3 | $140$ | $6$ | $6$ | $3$ |
| 140.192.5-28.k.1.1 | $140$ | $8$ | $8$ | $5$ |
| 140.240.7-140.s.1.8 | $140$ | $10$ | $10$ | $7$ |
| 140.504.16-28.s.1.1 | $140$ | $21$ | $21$ | $16$ |
| 280.48.0-56.be.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.be.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bf.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bf.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bm.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bm.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bn.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bn.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cm.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cm.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cn.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cn.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cu.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cu.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cv.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cv.1.15 | $280$ | $2$ | $2$ | $0$ |