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 $2$ are rational) | Cusp widths | $4^{6}\cdot20^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20M7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.288.7.37 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}7&6\\6&1\end{bmatrix}$, $\begin{bmatrix}7&7\\2&5\end{bmatrix}$, $\begin{bmatrix}19&19\\10&17\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $D_4\times F_5$ |
| Contains $-I$: | no $\quad$ (see 20.144.7.bd.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $16$ |
| Full 20-torsion field degree: | $160$ |
Jacobian
| Conductor: | $2^{28}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 80.2.a.a$^{2}$, 80.2.a.b, 80.2.c.a$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x u + y u - z v $ |
| $=$ | $x y + z^{2} + z t + w t + u^{2}$ | |
| $=$ | $x t - y z - y w + z u - w u - t u$ | |
| $=$ | $x^{2} + 3 x y + y^{2} - u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{10} y + x^{10} z - 4 x^{8} y^{3} + 11 x^{8} y^{2} z + 14 x^{8} y z^{2} + 3 x^{8} z^{3} + \cdots + 32 y^{5} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:1:0:1)$, $(0:0:0:-1:0:0:1)$ |
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.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x-3y+3z-2w-3t+3u+3v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x-4y+4z-w-4t-u-v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2x-2y+2z+2w-2t+2u+2v$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{3}Y-22X^{2}Y^{2}+6XY^{3}+12X^{2}YZ+14XY^{2}Z-6Y^{3}Z-3X^{2}Z^{2}-2XYZ^{2}+5Y^{2}Z^{2}-10YZ^{3}+2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.144.7.bd.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{10}Y+X^{10}Z-4X^{8}Y^{3}+11X^{8}Y^{2}Z+14X^{8}YZ^{2}+3X^{8}Z^{3}-38X^{6}Y^{5}-22X^{6}Y^{4}Z-4X^{6}Y^{3}Z^{2}+22X^{6}Y^{2}Z^{3}+12X^{6}YZ^{4}+X^{6}Z^{5}-56X^{4}Y^{7}-132X^{4}Y^{6}Z-88X^{4}Y^{5}Z^{2}+24X^{4}Y^{4}Z^{3}+2X^{4}Y^{3}Z^{4}+6X^{4}Y^{2}Z^{5}+2X^{4}YZ^{6}-32X^{2}Y^{9}-140X^{2}Y^{8}Z-8X^{2}Y^{7}Z^{2}+224X^{2}Y^{6}Z^{3}+120X^{2}Y^{5}Z^{4}+16X^{2}Y^{4}Z^{5}-8Y^{11}-24Y^{10}Z-24Y^{9}Z^{2}+32Y^{8}Z^{3}+96Y^{7}Z^{4}+96Y^{6}Z^{5}+32Y^{5}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.96.3-20.i.2.6 | $20$ | $3$ | $3$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 20.144.3-20.bm.1.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 20.144.3-20.bm.1.9 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
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.a.1.5 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 20.576.13-20.y.2.6 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 20.576.13-20.bi.2.1 | $20$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 20.576.13-20.bm.2.3 | $20$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 20.1440.43-20.cm.1.2 | $20$ | $5$ | $5$ | $43$ | $5$ | $1^{18}\cdot2^{9}$ |
| 40.576.13-40.gg.2.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 40.576.13-40.kq.2.13 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 40.576.13-40.uw.2.2 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2$ |
| 40.576.13-40.wi.2.9 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 40.576.17-40.bdm.1.1 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bdw.1.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bgu.1.2 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bha.1.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.buw.1.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bvg.1.1 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bwe.1.1 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.bwk.1.1 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{3}$ |
| 40.576.17-40.cmc.1.5 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cmi.1.5 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cng.1.5 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cnq.1.5 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cpu.1.1 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cqa.1.1 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.ctk.1.1 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.ctu.1.2 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 60.576.13-60.kb.2.8 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.kj.2.8 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.pi.2.6 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2$ |
| 60.576.13-60.pq.2.6 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2$ |
| 60.864.31-60.jp.1.2 | $60$ | $3$ | $3$ | $31$ | $7$ | $1^{10}\cdot2^{7}$ |
| 60.1152.37-60.hn.1.1 | $60$ | $4$ | $4$ | $37$ | $2$ | $1^{16}\cdot2^{7}$ |