Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $320$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.5.123 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}23&20\\4&39\end{bmatrix}$, $\begin{bmatrix}27&32\\3&21\end{bmatrix}$, $\begin{bmatrix}34&35\\5&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.5.b.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{24}\cdot5^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a, 320.2.a.a, 320.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} + x w - y z - 2 w^{2} + t^{2} $ |
| $=$ | $2 x^{2} - 4 x w + y^{2} + 2 y z + z^{2} - 2 w^{2}$ | |
| $=$ | $9 x^{2} - 3 x w - 2 y^{2} - 3 y z - 2 y t - 2 z^{2} + 2 z t + 6 w^{2} - 3 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} + 256 x^{7} z + 1440 x^{6} z^{2} + 3456 x^{5} z^{3} - 80 x^{4} y^{2} z^{2} + 3736 x^{4} z^{4} + \cdots + 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
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 20.48.3.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 3y+2z+3t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y+z-t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2y-2z+2t$ |
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 40.96.5.b.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x-w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{4}{5}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y+z+2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{8}+256X^{7}Z+1440X^{6}Z^{2}-80X^{4}Y^{2}Z^{2}+3456X^{5}Z^{3}+3736X^{4}Z^{4}-400X^{2}Y^{2}Z^{4}+500Y^{4}Z^{4}+1728X^{3}Z^{5}+360X^{2}Z^{6}-20Y^{2}Z^{6}+32XZ^{7}+Z^{8} $ |
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 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.48.1-40.bx.2.1 | $40$ | $4$ | $4$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.96.3-20.i.2.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.13-40.gg.2.2 | $40$ | $3$ | $3$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.768.25-40.c.2.6 | $40$ | $4$ | $4$ | $25$ | $2$ | $1^{8}\cdot2^{4}\cdot4$ |
| 40.960.29-40.bar.1.2 | $40$ | $5$ | $5$ | $29$ | $6$ | $1^{12}\cdot2^{6}$ |
| 80.384.13-80.i.1.3 | $80$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 80.384.13-80.m.2.5 | $80$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 80.384.13-80.q.2.3 | $80$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 80.384.13-80.u.1.6 | $80$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.13-240.bbb.2.13 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.13-240.bbf.1.11 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.13-240.bbz.1.14 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.384.13-240.bcd.2.11 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |