Invariants
Level: | $280$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
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: | 8G0 |
Level structure
$\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}25&228\\116&271\end{bmatrix}$, $\begin{bmatrix}27&140\\26&249\end{bmatrix}$, $\begin{bmatrix}103&72\\172&121\end{bmatrix}$, $\begin{bmatrix}145&36\\98&109\end{bmatrix}$, $\begin{bmatrix}209&156\\270&243\end{bmatrix}$, $\begin{bmatrix}239&76\\26&87\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.24.0.m.1 for the level structure with $-I$) |
Cyclic 280-isogeny field degree: | $96$ |
Cyclic 280-torsion field degree: | $9216$ |
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 44 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{5^4}\cdot\frac{(10x-3y)^{24}(96160000x^{8}-242560000x^{7}y+254560000x^{6}y^{2}-148768000x^{5}y^{3}+57141600x^{4}y^{4}-13524800x^{3}y^{5}+2143600x^{2}y^{6}-154160xy^{7}+5161y^{8})^{3}}{(2x+y)^{8}(10x-3y)^{32}(20x^{2}-20xy+y^{2})^{2}(60x^{2}-20xy+7y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
56.24.0-4.b.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
140.24.0-4.b.1.3 | $140$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-40.z.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-40.z.1.14 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-40.ba.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-40.ba.1.14 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.