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: | $\begin{bmatrix}23&50\\26&31\end{bmatrix}$, $\begin{bmatrix}29&50\\122&113\end{bmatrix}$, $\begin{bmatrix}91&8\\24&125\end{bmatrix}$, $\begin{bmatrix}107&58\\104&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.12.0.a.1 for the level structure with $-I$) |
| Cyclic 140-isogeny field degree: | $96$ |
| Cyclic 140-torsion field degree: | $4608$ |
| 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 480 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^2}\cdot\frac{x^{12}(25x^{4}+1280x^{2}y^{2}+65536y^{4})^{3}}{y^{4}x^{16}(5x^{2}+256y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-2.a.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 140.12.0-2.a.1.1 | $140$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 140.48.0-20.a.1.1 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-20.a.1.2 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-20.c.1.3 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-20.c.1.4 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-140.d.1.4 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-140.d.1.6 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-140.e.1.4 | $140$ | $2$ | $2$ | $0$ |
| 140.48.0-140.e.1.8 | $140$ | $2$ | $2$ | $0$ |
| 140.120.4-20.c.1.1 | $140$ | $5$ | $5$ | $4$ |
| 140.144.3-20.c.1.3 | $140$ | $6$ | $6$ | $3$ |
| 140.192.5-140.a.1.1 | $140$ | $8$ | $8$ | $5$ |
| 140.240.7-20.c.1.2 | $140$ | $10$ | $10$ | $7$ |
| 140.504.16-140.a.1.4 | $140$ | $21$ | $21$ | $16$ |
| 280.48.0-40.b.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.b.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.f.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.f.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.h.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.h.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.k.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.k.1.11 | $280$ | $2$ | $2$ | $0$ |