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: | $\begin{bmatrix}9&172\\209&165\end{bmatrix}$, $\begin{bmatrix}61&112\\195&251\end{bmatrix}$, $\begin{bmatrix}75&116\\31&147\end{bmatrix}$, $\begin{bmatrix}101&12\\271&149\end{bmatrix}$, $\begin{bmatrix}143&20\\183&67\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bh.1 for the level structure with $-I$) |
Cyclic 280-isogeny field degree: | $48$ |
Cyclic 280-torsion field degree: | $4608$ |
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 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2}{7}\cdot\frac{(x+y)^{24}(2401x^{8}+41160x^{6}y^{2}+26264x^{4}y^{4}+3360x^{2}y^{6}+16y^{8})^{3}}{y^{2}x^{2}(x+y)^{24}(7x^{2}-2y^{2})^{8}(7x^{2}+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
40.24.0-8.n.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
280.24.0-8.n.1.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.s.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.s.1.7 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.bb.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
280.24.0-56.bb.1.8 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
280.96.0-56.bi.1.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.bi.2.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.bj.1.2 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.bj.2.2 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.dq.1.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.dq.2.13 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.dr.1.7 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.dr.2.9 | $280$ | $2$ | $2$ | $0$ |
280.240.8-280.dt.1.17 | $280$ | $5$ | $5$ | $8$ |
280.288.7-280.he.1.26 | $280$ | $6$ | $6$ | $7$ |
280.384.11-56.dx.1.17 | $280$ | $8$ | $8$ | $11$ |
280.480.15-280.jf.1.41 | $280$ | $10$ | $10$ | $15$ |