Invariants
Level: | $280$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 4E0 |
Level structure
$\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}47&20\\70&263\end{bmatrix}$, $\begin{bmatrix}59&216\\90&23\end{bmatrix}$, $\begin{bmatrix}117&188\\69&223\end{bmatrix}$, $\begin{bmatrix}187&36\\198&15\end{bmatrix}$, $\begin{bmatrix}219&48\\222&157\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.12.0.s.1 for the level structure with $-I$) |
Cyclic 280-isogeny field degree: | $96$ |
Cyclic 280-torsion field degree: | $9216$ |
Full 280-torsion field degree: | $61931520$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 317 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{5\cdot7^4}\cdot\frac{(2x+5y)^{12}(444x^{4}+1920x^{3}y-2740x^{2}y^{2}-4800xy^{3}+2775y^{4})^{3}}{(x-y)^{2}(2x+5y)^{14}(2x^{2}+5y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
28.12.0-4.c.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
280.12.0-4.c.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
280.48.0-40.bk.1.3 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bk.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bl.1.4 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bl.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bu.1.3 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bu.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bv.1.4 | $280$ | $2$ | $2$ | $0$ |
280.48.0-40.bv.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.by.1.8 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.by.1.9 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.bz.1.7 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.bz.1.10 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.cc.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.cc.1.11 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.cd.1.3 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.cd.1.14 | $280$ | $2$ | $2$ | $0$ |
280.120.4-40.be.1.2 | $280$ | $5$ | $5$ | $4$ |
280.144.3-40.bq.1.6 | $280$ | $6$ | $6$ | $3$ |
280.192.5-280.be.1.2 | $280$ | $8$ | $8$ | $5$ |
280.240.7-40.cc.1.4 | $280$ | $10$ | $10$ | $7$ |
280.504.16-280.cc.1.4 | $280$ | $21$ | $21$ | $16$ |