Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.288.5.65 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}11&0\\6&13\end{bmatrix}$, $\begin{bmatrix}13&3\\2&3\end{bmatrix}$, $\begin{bmatrix}15&7\\14&17\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_2^3:F_5$ |
| Contains $-I$: | no $\quad$ (see 20.144.5.be.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $8$ |
| Full 20-torsion field degree: | $160$ |
Jacobian
| Conductor: | $2^{18}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a, 100.2.a.a, 400.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $2 x y - 5 x z + 2 y^{2} + 3 z^{2} - w^{2}$ | |
| $=$ | $5 x^{2} + 3 x y + 10 x z + 3 y^{2} + 7 z^{2} - 2 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{8} - 26 x^{7} y - 9 x^{6} y^{2} + 270 x^{6} z^{2} + 4 x^{5} y^{3} - 160 x^{5} y z^{2} + \cdots + 625 z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{79994880z^{2}w^{16}-89763840z^{2}w^{14}t^{2}+33085440z^{2}w^{12}t^{4}+16542720z^{2}w^{10}t^{6}-38822400z^{2}w^{8}t^{8}+27774720z^{2}w^{6}t^{10}-9214560z^{2}w^{4}t^{12}+1406160z^{2}w^{2}t^{14}-78120z^{2}t^{16}-3198976w^{18}+2752512w^{16}t^{2}-55296w^{14}t^{4}-4791040w^{12}t^{6}+9047040w^{10}t^{8}-7471104w^{8}t^{10}+3137488w^{6}t^{12}-690624w^{4}t^{14}+75000w^{2}t^{16}-3125t^{18}}{t^{2}w^{4}(320z^{2}w^{10}+200z^{2}w^{8}t^{2}+100z^{2}w^{6}t^{4}+50z^{2}w^{4}t^{6}-50z^{2}w^{2}t^{8}+5z^{2}t^{10}-64w^{12}-24w^{10}t^{2}-9w^{8}t^{4}-4w^{6}t^{6}+w^{4}t^{8})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.144.5.be.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y+2w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 11X^{8}-26X^{7}Y-9X^{6}Y^{2}+270X^{6}Z^{2}+4X^{5}Y^{3}-160X^{5}YZ^{2}+X^{4}Y^{4}-80X^{4}Y^{2}Z^{2}+1400X^{4}Z^{4}-150X^{3}YZ^{4}-75X^{2}Y^{2}Z^{4}+1750X^{2}Z^{6}+625Z^{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.i.1.1 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-20.i.1.4 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-20.m.2.5 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.1-20.m.2.7 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.144.3-20.bm.2.9 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.144.3-20.bm.2.11 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.576.13-20.bm.1.3 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
| 20.576.13-20.bn.1.3 | $20$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 20.1440.37-20.ca.1.2 | $20$ | $5$ | $5$ | $37$ | $3$ | $1^{16}\cdot2^{8}$ |
| 40.576.13-40.tu.1.11 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.tw.1.9 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.wc.1.10 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.wd.1.10 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.wq.1.7 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.576.13-40.ws.1.5 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.576.17-40.bbq.2.11 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}\cdot4$ |
| 40.576.17-40.bbu.2.13 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 40.576.17-40.ciy.2.13 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}\cdot4$ |
| 40.576.17-40.cjc.2.11 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| 60.576.13-60.ow.1.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.576.13-60.ox.1.1 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.864.29-60.dym.2.17 | $60$ | $3$ | $3$ | $29$ | $4$ | $1^{12}\cdot2^{6}$ |
| 60.1152.33-60.or.2.11 | $60$ | $4$ | $4$ | $33$ | $2$ | $1^{14}\cdot2^{7}$ |