Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $6$ are rational) | Cusp widths | $4^{6}\cdot20^{6}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.288.7.5 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}7&12\\18&17\end{bmatrix}$, $\begin{bmatrix}11&0\\8&7\end{bmatrix}$, $\begin{bmatrix}11&10\\6&3\end{bmatrix}$, $\begin{bmatrix}15&8\\18&7\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_2^3\times F_5$ |
| Contains $-I$: | no $\quad$ (see 20.144.7.f.2 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: | $2^{26}\cdot5^{13}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2\cdot4$ |
| Newforms: | 20.2.a.a, 400.2.n.a, 400.2.n.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ w v - t u $ |
| $=$ | $y v + z u$ | |
| $=$ | $y v - z v - u^{2} + u v$ | |
| $=$ | $z v - t^{2} + t u - t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 25 x^{4} y^{5} - 75 x^{4} y^{4} z - 75 x^{4} y^{3} z^{2} - 25 x^{4} y^{2} z^{3} + y^{7} z^{2} + \cdots + z^{9} $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:1)$, $(0:1:0:1:0:0:0)$, $(0:1:0:0:0:0:0)$, $(0:0:0:0:-1:0:1)$, $(0:1:1:0:0:0:0)$, $(0:0:0:0:0:1:1)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 20.48.3.a.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 5x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-3v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-2v$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2Y^{4}-3Y^{3}Z-4Y^{2}Z^{2}+4YZ^{3}+3Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.144.7.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -25X^{4}Y^{5}-75X^{4}Y^{4}Z-75X^{4}Y^{3}Z^{2}-25X^{4}Y^{2}Z^{3}+Y^{7}Z^{2}+Y^{6}Z^{3}-12Y^{5}Z^{4}+10Y^{4}Z^{5}+10Y^{3}Z^{6}-8Y^{2}Z^{7}-3YZ^{8}+Z^{9} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $2$ | $2$ | $1$ | $0$ | $2\cdot4$ |
| 20.96.3-20.a.2.1 | $20$ | $3$ | $3$ | $3$ | $0$ | $4$ |
| 20.144.1-10.a.1.10 | $20$ | $2$ | $2$ | $1$ | $0$ | $2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.576.13-20.k.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.l.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.m.2.8 | $20$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.n.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.1440.43-20.i.1.7 | $20$ | $5$ | $5$ | $43$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{3}\cdot8$ |
| 40.576.13-40.by.2.15 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.cb.1.15 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.ce.1.15 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.ch.2.15 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.ek.1.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.el.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.eq.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.er.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.864.31-60.p.2.32 | $60$ | $3$ | $3$ | $31$ | $0$ | $1^{6}\cdot2^{5}\cdot8$ |
| 60.1152.37-60.p.2.30 | $60$ | $4$ | $4$ | $37$ | $0$ | $1^{6}\cdot2^{4}\cdot4^{2}\cdot8$ |