Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
| 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: | 20.288.7.4 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}1&10\\14&19\end{bmatrix}$, $\begin{bmatrix}17&2\\2&5\end{bmatrix}$, $\begin{bmatrix}19&6\\0&1\end{bmatrix}$, $\begin{bmatrix}19&6\\12&15\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_2^3\times F_5$ |
| Contains $-I$: | no $\quad$ (see 20.144.7.g.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: | $2^{26}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2\cdot4$ |
| Newforms: | 20.2.a.a, 80.2.n.a, 80.2.n.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x v - t u $ |
| $=$ | $x v - z w$ | |
| $=$ | $x v + t v - u^{2} + u v$ | |
| $=$ | $x t - x u + x v + t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y^{4} + 2 x^{4} y^{2} z^{2} - x^{4} z^{4} - y^{7} z - 7 y^{6} z^{2} - 5 y^{5} z^{3} + \cdots + y z^{7} $ |
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 |
|---|
| $(0:0:0:0:0:1:1)$, $(-1:0:0:0:1:0:0)$, $(1:0:0:0:0:0:0)$, $(0:0:0:0:0:0:1)$, $(-1:0:0:1:0:0:0)$, $(0:0:1:0:0:0:1)$ |
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.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-11X^{2}Y^{2}-XY^{3}-Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.144.7.g.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{4}Y^{4}+2X^{4}Y^{2}Z^{2}-X^{4}Z^{4}-Y^{7}Z-7Y^{6}Z^{2}-5Y^{5}Z^{3}+30Y^{4}Z^{4}+5Y^{3}Z^{5}-7Y^{2}Z^{6}+YZ^{7} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.12.0-2.a.1.2 | $4$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
| $X_1(5)$ | $5$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
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.c.1.3 | $20$ | $3$ | $3$ | $3$ | $0$ | $4$ |
| 20.144.1-10.a.1.9 | $20$ | $2$ | $2$ | $1$ | $0$ | $2\cdot4$ |
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.o.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.p.1.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.q.1.8 | $20$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 20.576.13-20.r.2.8 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 20.1440.43-20.p.1.8 | $20$ | $5$ | $5$ | $43$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{3}\cdot8$ |
| 40.576.13-40.ck.2.14 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.cn.2.15 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.cq.2.15 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{2}\cdot2^{2}$ |
| 40.576.13-40.ct.2.14 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.ew.2.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.ex.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fc.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.576.13-60.fd.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 60.864.31-60.r.1.32 | $60$ | $3$ | $3$ | $31$ | $0$ | $1^{6}\cdot2^{5}\cdot8$ |
| 60.1152.37-60.r.1.30 | $60$ | $4$ | $4$ | $37$ | $0$ | $1^{6}\cdot2^{4}\cdot4^{2}\cdot8$ |