Invariants
Level: | $280$ | $\SL_2$-level: | $4$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4G0 |
Level structure
$\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}51&66\\110&149\end{bmatrix}$, $\begin{bmatrix}197&186\\90&149\end{bmatrix}$, $\begin{bmatrix}239&262\\16&189\end{bmatrix}$, $\begin{bmatrix}279&92\\278&207\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.d.1 for the level structure with $-I$) |
Cyclic 280-isogeny field degree: | $192$ |
Cyclic 280-torsion field degree: | $18432$ |
Full 280-torsion field degree: | $30965760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 26 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{7^2}\cdot\frac{(4x+y)^{24}(3735552x^{8}-2752512x^{7}y+5619712x^{6}y^{2}+8429568x^{5}y^{3}+5092864x^{4}y^{4}-3687936x^{3}y^{5}+1075648x^{2}y^{6}+230496xy^{7}+136857y^{8})^{3}}{(4x+y)^{24}(16x^{2}+7y^{2})^{4}(16x^{2}-8xy-7y^{2})^{4}(16x^{2}+56xy-7y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
40.24.0-8.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
140.24.0-28.b.1.3 | $140$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-8.a.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-28.b.1.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.b.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.b.1.7 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
280.240.8-280.d.1.13 | $280$ | $5$ | $5$ | $8$ |
280.288.7-280.f.1.10 | $280$ | $6$ | $6$ | $7$ |
280.384.11-56.g.1.6 | $280$ | $8$ | $8$ | $11$ |
280.480.15-280.d.1.27 | $280$ | $10$ | $10$ | $15$ |