Invariants
| Level: | $280$ | $\SL_2$-level: | $20$ | 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 | $10^{3}$ | 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: | 10B2 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}12&161\\53&232\end{bmatrix}$, $\begin{bmatrix}38&203\\91&72\end{bmatrix}$, $\begin{bmatrix}61&142\\20&277\end{bmatrix}$, $\begin{bmatrix}71&256\\182&279\end{bmatrix}$, $\begin{bmatrix}162&245\\41&48\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.30.2.a.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 $ | $=$ | $ 3 x^{2} w + x w^{2} - y^{2} w - y z w $ |
| $=$ | $3 x^{2} z + x z w - y^{2} z - y z^{2}$ | |
| $=$ | $2 x^{2} y - x^{2} z - 2 x y w + 2 y^{3} + y^{2} z + z^{3} + z w^{2}$ | |
| $=$ | $3 x^{2} y + x y w - y^{3} - y^{2} z$ | |
| $=$ | $\cdots$ |
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 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:0:1)$ | $(1:0:0)$ | $(0:-1:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(3z^{2}-w^{2})^{3}}{z^{2}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.30.2.a.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.a.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.a.1.1 | $56$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.a.1.1 | $56$ | $5$ | $5$ | $0$ | $0$ |
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.e.1.1 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.e.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.f.1.1 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.f.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.g.1.1 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.g.1.1 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-20.h.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-140.h.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.m.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.m.1.5 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.p.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.p.1.5 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.s.1.4 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.s.1.2 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-40.v.1.3 | $280$ | $2$ | $2$ | $4$ |
| 280.120.4-280.v.1.7 | $280$ | $2$ | $2$ | $4$ |
| 280.180.4-20.a.1.1 | $280$ | $3$ | $3$ | $4$ |
| 280.240.5-20.d.1.2 | $280$ | $4$ | $4$ | $5$ |
| 280.480.18-140.a.1.3 | $280$ | $8$ | $8$ | $18$ |