Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}35&152\\66&107\end{bmatrix}$, $\begin{bmatrix}141&220\\96&201\end{bmatrix}$, $\begin{bmatrix}177&132\\120&267\end{bmatrix}$, $\begin{bmatrix}201&260\\146&181\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.1.bw.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| Cyclic 280-torsion field degree: | $9216$ |
| Full 280-torsion field degree: | $7741440$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - y^{2} - z^{2} $ |
| $=$ | $4 x^{2} + 2 y^{2} + 2 y w + 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 12 x^{2} y^{2} + 16 x^{2} z^{2} + 4 y^{4} + 6 y^{2} z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2681928yz^{22}w-211788yz^{20}w^{3}-17582868yz^{18}w^{5}-13119534yz^{16}w^{7}+20892960yz^{14}w^{9}+42601104yz^{12}w^{11}+34492752yz^{10}w^{13}+16270488yz^{8}w^{15}+4822200yz^{6}w^{17}+888300yz^{4}w^{19}+93060yz^{2}w^{21}+4230yw^{23}+7189057z^{24}-27626406z^{22}w^{2}-1397817z^{20}w^{4}+54268255z^{18}w^{6}+36194361z^{16}w^{8}-23807256z^{14}w^{10}-47705476z^{12}w^{12}-32902140z^{10}w^{14}-13174095z^{8}w^{16}-3341530z^{6}w^{18}-531783z^{4}w^{20}-48531z^{2}w^{22}-1934w^{24}}{z^{8}(3024yz^{14}w+19464yz^{12}w^{3}+49176yz^{10}w^{5}+61764yz^{8}w^{7}+40920yz^{6}w^{9}+14508yz^{4}w^{11}+2604yz^{2}w^{13}+186yw^{15}-3969z^{16}-29484z^{14}w^{2}-90406z^{12}w^{4}-146506z^{10}w^{6}-133721z^{8}w^{8}-68794z^{6}w^{10}-19721z^{4}w^{12}-2941z^{2}w^{14}-178w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.1.bw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+12X^{2}Y^{2}+4Y^{4}+16X^{2}Z^{2}+6Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.96.0-8.h.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 280.96.0-8.h.1.8 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.h.2.1 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.h.2.11 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.j.2.6 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.j.2.13 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.y.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.0-56.y.1.10 | $280$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 280.96.1-56.bh.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.bh.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.bj.2.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.bj.2.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.bs.1.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.bs.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |