Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{2}\cdot4^{3}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.384.5.138 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&4\\4&3\end{bmatrix}$, $\begin{bmatrix}21&4\\4&33\end{bmatrix}$, $\begin{bmatrix}31&12\\0&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.192.5.e.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $1920$ |
Jacobian
| Conductor: | $2^{28}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 64.2.a.a, 800.2.a.d$^{2}$, 1600.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} + z w $ |
| $=$ | $z^{2} + w^{2} + t^{2}$ | |
| $=$ | $5 x^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{8} - 32 x^{7} z + 136 x^{6} z^{2} - 320 x^{5} z^{3} + 25 x^{4} y^{4} + 520 x^{4} z^{4} + \cdots + 48 z^{8} $ |
Rational points
This modular curve has no real points, and therefore no 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 40.96.3.v.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x-t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6Y^{4}-2Y^{3}Z-6Y^{2}Z^{2}-8YZ^{3}-4Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.192.5.e.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y-t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z-\frac{1}{2}w-\frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{8}+25X^{4}Y^{4}-32X^{7}Z-100X^{3}Y^{4}Z+136X^{6}Z^{2}+150X^{2}Y^{4}Z^{2}-320X^{5}Z^{3}-100XY^{4}Z^{3}+520X^{4}Z^{4}+25Y^{4}Z^{4}-640X^{3}Z^{5}+544X^{2}Z^{6}-256XZ^{7}+48Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.1-8.a.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-8.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.b.2.2 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.1-40.b.2.11 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.1-40.o.2.6 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.1-40.o.2.11 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.3-40.m.1.8 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.192.3-40.m.1.11 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.192.3-40.u.2.11 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.2.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.v.1.8 | $40$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 40.192.3-40.v.1.14 | $40$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 40.192.3-40.z.2.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.192.3-40.z.2.12 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.69-40.bd.2.2 | $40$ | $5$ | $5$ | $69$ | $17$ | $1^{26}\cdot2^{15}\cdot4^{2}$ |
| 40.2304.73-40.lh.2.4 | $40$ | $6$ | $6$ | $73$ | $10$ | $1^{28}\cdot2^{4}\cdot4^{8}$ |
| 40.3840.137-40.hu.1.5 | $40$ | $10$ | $10$ | $137$ | $24$ | $1^{54}\cdot2^{19}\cdot4^{10}$ |