Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
| 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 | $1^{4}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-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: | 20.288.5.4 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}9&16\\14&9\end{bmatrix}$, $\begin{bmatrix}11&0\\4&9\end{bmatrix}$, $\begin{bmatrix}15&8\\12&15\end{bmatrix}$, $\begin{bmatrix}19&6\\16&3\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_2^3\times F_5$ |
| Contains $-I$: | no $\quad$ (see 20.144.5.f.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $4$ |
| Full 20-torsion field degree: | $160$ |
Jacobian
| Conductor: | $2^{13}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 20.2.a.a, 40.2.c.a, 100.2.a.a, 200.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y w - y t - w^{2} $ |
| $=$ | $y z + z^{2} - t^{2}$ | |
| $=$ | $5 x^{2} - y t - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} - x^{4} z^{2} + 20 x^{2} y^{2} z^{2} - 125 y^{4} z^{2} + 25 y^{2} z^{4} $ |
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 |
|---|
| $(0:1:0:0:0)$, $(0:1:-1:1:0)$, $(0:-1:1:0:0)$, $(0:1:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{6}-4y^{5}t+16yt^{5}+16t^{6})^{3}}{t^{10}y^{5}(y-4t)(y+t)^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.144.5.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}y+\frac{1}{5}z-\frac{1}{5}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{2}-X^{4}Z^{2}+20X^{2}Y^{2}Z^{2}-125Y^{4}Z^{2}+25Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-10.a.1.7 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-20.m.2.2 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-20.m.2.7 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.3-20.bg.2.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.144.3-20.bg.2.7 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.576.9-20.f.1.2 | $20$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 20.576.9-20.f.3.4 | $20$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 20.576.13-20.g.2.6 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
| 20.576.13-20.h.1.8 | $20$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 20.576.13-20.n.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 20.576.13-20.r.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 20.1440.37-20.e.1.6 | $20$ | $5$ | $5$ | $37$ | $1$ | $1^{16}\cdot2^{8}$ |
| 40.576.9-40.l.1.4 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 40.576.9-40.l.3.7 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 40.576.13-40.bc.1.13 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.bi.1.13 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.ci.2.12 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 40.576.13-40.cu.2.14 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 60.576.9-60.r.1.4 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.576.9-60.r.3.8 | $60$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 60.576.13-60.cw.1.15 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.cy.1.15 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.et.2.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 60.576.13-60.ff.1.15 | $60$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 60.864.29-60.r.2.3 | $60$ | $3$ | $3$ | $29$ | $2$ | $1^{12}\cdot2^{6}$ |
| 60.1152.33-60.bd.2.7 | $60$ | $4$ | $4$ | $33$ | $1$ | $1^{14}\cdot2^{7}$ |