Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | $\begin{bmatrix}5&8\\124&17\end{bmatrix}$, $\begin{bmatrix}79&192\\56&131\end{bmatrix}$, $\begin{bmatrix}111&236\\10&163\end{bmatrix}$, $\begin{bmatrix}137&76\\32&239\end{bmatrix}$, $\begin{bmatrix}205&104\\262&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.r.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: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 14 x y + w^{2} $ |
| $=$ | $14 x^{2} - 14 y^{2} - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 14 x^{2} y^{2} - 196 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{4}+w^{4})^{3}}{w^{8}z^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.48.1.r.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{14}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+14X^{2}Y^{2}-196Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.1-8.c.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 280.48.1-8.c.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.0-56.e.1.8 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.0-56.e.1.15 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.0-56.l.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.48.0-56.l.1.16 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 280.192.1-56.d.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.d.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.j.1.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.j.2.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.u.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.u.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.z.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.z.2.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.l.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.l.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bg.1.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bg.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.cf.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.cf.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.cx.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.cx.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.480.17-280.bj.1.17 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |