Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}23&200\\80&179\end{bmatrix}$, $\begin{bmatrix}159&76\\51&185\end{bmatrix}$, $\begin{bmatrix}185&96\\233&171\end{bmatrix}$, $\begin{bmatrix}275&172\\191&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.ea.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| Cyclic 280-torsion field degree: | $9216$ |
| Full 280-torsion field degree: | $15482880$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} + 2 x y - 4 x z + y^{2} $ |
| $=$ | $5 y z - 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} - 10 x^{3} y + 2 x^{2} y^{2} - 20 x^{2} z^{2} - 4 x y z^{2} + 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^2}\cdot\frac{(5y^{2}-4w^{2})^{3}(5y^{2}+4w^{2})^{3}}{w^{8}y^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.ea.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 5x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{4}-10X^{3}Y+2X^{2}Y^{2}-20X^{2}Z^{2}-4XYZ^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.48.0-4.c.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 280.48.0-4.c.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.0-40.bf.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.0-40.bf.1.6 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.1-40.n.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1-40.n.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 280.480.17-40.hw.1.4 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |