Invariants
| Level: | $280$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $60$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $2 = 1 + \frac{ 30 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $5^{2}\cdot20$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A2 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}87&246\\18&183\end{bmatrix}$, $\begin{bmatrix}108&77\\1&206\end{bmatrix}$, $\begin{bmatrix}121&94\\158&189\end{bmatrix}$, $\begin{bmatrix}177&174\\94&55\end{bmatrix}$, $\begin{bmatrix}256&19\\33&254\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.30.2.b.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $192$ |
| Cyclic 280-torsion field degree: | $18432$ |
| Full 280-torsion field degree: | $24772608$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{3} + 2 x^{2} w - 2 x y^{2} - 2 x z^{2} + y z w $ |
| $=$ | $3 x^{2} w + x w^{2} - y^{2} w + 2 y z w$ | |
| $=$ | $x^{3} - x^{2} w + x y^{2} + 2 x y z + 2 x z^{2} - y z w$ | |
| $=$ | $3 x^{2} z + x z w - y^{2} z + 2 y z^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{5} + 6 x^{4} y + x^{3} y^{2} - 10 x^{3} z^{2} - 3 x^{2} y z^{2} + x y^{2} z^{2} + 5 x z^{4} - y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x\right) y $ | $=$ | $ -4x^{4} + 4x^{2} - 8 $ |
Rational points
This modular curve has 1 rational cusp but no known non-cuspidal rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | |||||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:0:1)$ | $(1:0:0)$ | $(0:2:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(3z^{2}+w^{2})^{3}}{z^{4}(4z^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.30.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{5}+6X^{4}Y+X^{3}Y^{2}-10X^{3}Z^{2}-3X^{2}YZ^{2}+XY^{2}Z^{2}+5XZ^{4}-YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.30.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -3x^{4}y^{2}-x^{3}y^{2}w+x^{2}y^{4}-xy^{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -xy$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $12$ | $6$ | $0$ | $0$ |
| 56.12.0-4.b.1.3 | $56$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $5$ | $5$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.120.4-20.a.1.10 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.e.1.6 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.e.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.i.1.7 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.i.1.5 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.j.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.j.1.6 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.m.1.5 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.n.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.n.1.5 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.y.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.y.1.8 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.bb.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.bb.1.7 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.bk.1.7 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.bn.1.7 | $280$ | $2$ | $2$ | $4$ |
| 280.180.4-20.b.1.12 | $280$ | $3$ | $3$ | $4$ |
| 280.240.5-20.k.1.3 | $280$ | $4$ | $4$ | $5$ |
| 280.480.18-140.b.1.11 | $280$ | $8$ | $8$ | $18$ |