Invariants
| Level: | $280$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot14^{2}\cdot28^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28E5 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}65&154\\78&227\end{bmatrix}$, $\begin{bmatrix}117&28\\22&9\end{bmatrix}$, $\begin{bmatrix}193&182\\57&221\end{bmatrix}$, $\begin{bmatrix}221&238\\30&179\end{bmatrix}$, $\begin{bmatrix}237&98\\214&227\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.5.y.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $24$ |
| Cyclic 280-torsion field degree: | $2304$ |
| Full 280-torsion field degree: | $7741440$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z + x w - y t $ |
| $=$ | $14 x^{2} + z w$ | |
| $=$ | $14 y^{2} - z^{2} + 5 z w - w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 14 x^{6} - 392 x^{4} y^{2} - x^{4} z^{2} + 2744 x^{2} y^{4} - 70 x^{2} y^{2} z^{2} - 196 y^{4} z^{2} - y^{2} z^{4} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{570391865302xyw^{9}t+24455499097372xyw^{7}t^{3}+200603549368922xyw^{5}t^{5}+492349347836444xyw^{3}t^{7}+385910721293846xywt^{9}-7529536z^{12}-135531648z^{10}t^{2}-1784500032z^{8}t^{4}-22453076352z^{6}t^{6}-286920498816z^{4}t^{8}-3762313371264z^{2}t^{10}+5679882025zw^{11}+876962131284zw^{9}t^{2}+14294152376275zw^{7}t^{4}+58637349395624zw^{5}t^{6}+76132571004277zw^{3}t^{8}+19495956502242zwt^{10}-64w^{12}-35062395220w^{10}t^{2}-1109616170513w^{8}t^{4}-7119342421398w^{6}t^{6}-15613300332207w^{4}t^{8}-13064701019398w^{2}t^{10}-3496189450881t^{12}}{w(5488xyw^{8}t-3822xyw^{6}t^{3}-2394xyw^{4}t^{5}-266xyw^{2}t^{7}-14xyt^{9}+343zw^{10}-539zw^{8}t^{2}+790zw^{6}t^{4}+218zw^{4}t^{6}+19zw^{2}t^{8}+zt^{10}-49w^{9}t^{2}-330w^{7}t^{4}-360w^{5}t^{6}-86w^{3}t^{8}-7wt^{10})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.5.y.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{14}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 14X^{6}-392X^{4}Y^{2}+2744X^{2}Y^{4}-X^{4}Z^{2}-70X^{2}Y^{2}Z^{2}-196Y^{4}Z^{2}-Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 280.24.0-56.m.1.3 | $280$ | $8$ | $8$ | $0$ | $?$ |
| 280.96.2-28.b.1.15 | $280$ | $2$ | $2$ | $2$ | $?$ |
| 280.96.2-28.b.1.16 | $280$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.384.11-56.ee.1.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ee.1.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ee.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ee.2.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ef.1.1 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ef.1.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ef.2.1 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ef.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jg.1.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jg.1.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jg.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jg.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jh.1.9 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jh.1.11 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jh.2.9 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.jh.2.11 | $280$ | $2$ | $2$ | $11$ |