Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.5.175 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&23\\32&17\end{bmatrix}$, $\begin{bmatrix}19&15\\36&23\end{bmatrix}$, $\begin{bmatrix}37&7\\0&19\end{bmatrix}$, $\begin{bmatrix}37&34\\28&3\end{bmatrix}$, $\begin{bmatrix}39&32\\16&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.144.5.fz.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $16$ |
| Full 40-torsion field degree: | $2560$ |
Jacobian
| Conductor: | $2^{26}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 20.2.a.a, 320.2.c.c, 1600.2.a.c, 1600.2.a.o |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y z - z^{2} + w^{2} $ |
| $=$ | $y w + w^{2} - t^{2}$ | |
| $=$ | $10 x^{2} - y^{2} + 3 y w - w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 10000 x^{8} - 4000 x^{6} z^{2} - 600 x^{4} y^{2} z^{2} + 600 x^{4} z^{4} - 40 x^{2} y^{2} z^{4} + \cdots + z^{8} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:-1:1)$, $(0:0:-1:1:1)$, $(0:0:1:-1:1)$, $(0:0:1:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{y^{18}-12y^{16}t^{2}+60y^{14}t^{4}-184y^{12}t^{6}+492y^{10}t^{8}-1368y^{8}t^{10}+3736y^{6}t^{12}-10320y^{4}t^{14}+30156y^{2}t^{16}+124w^{18}-720w^{16}t^{2}+5220w^{14}t^{4}-18720w^{12}t^{6}+64560w^{10}t^{8}-148320w^{8}t^{10}+281360w^{6}t^{12}-381600w^{4}t^{14}+294600w^{2}t^{16}-92408t^{18}}{t^{4}w^{2}(w-t)^{5}(w+t)^{5}(5w^{2}-t^{2})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.5.fz.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y-2z+2t$ |
Equation of the image curve:
| $0$ | $=$ | $ 10000X^{8}-4000X^{6}Z^{2}-600X^{4}Y^{2}Z^{2}+600X^{4}Z^{4}-40X^{2}Y^{2}Z^{4}+5Y^{4}Z^{4}-40X^{2}Z^{6}-6Y^{2}Z^{6}+Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.144.1-20.f.2.11 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.144.1-20.f.2.15 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.144.1-40.bf.1.3 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.144.1-40.bf.1.8 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.144.3-40.ek.2.5 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.ek.2.14 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.9-40.be.1.7 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{2}$ |
| 40.576.9-40.be.3.6 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{2}$ |
| 40.576.9-40.bf.2.7 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{2}$ |
| 40.576.9-40.bf.4.7 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{2}$ |
| 40.576.13-40.no.2.15 | $40$ | $2$ | $2$ | $13$ | $2$ | $2^{2}\cdot4$ |
| 40.576.13-40.np.2.15 | $40$ | $2$ | $2$ | $13$ | $2$ | $2^{2}\cdot4$ |
| 40.576.13-40.ok.2.15 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.ol.1.22 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.qc.1.14 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.qd.2.15 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.rc.1.13 | $40$ | $2$ | $2$ | $13$ | $2$ | $2^{2}\cdot4$ |
| 40.576.13-40.rd.1.13 | $40$ | $2$ | $2$ | $13$ | $2$ | $2^{2}\cdot4$ |
| 40.1440.37-40.ky.1.2 | $40$ | $5$ | $5$ | $37$ | $7$ | $1^{16}\cdot2^{8}$ |