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 | $4^{4}\cdot8$ | ||
| 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: | 8A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.384.5.423 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&0\\0&19\end{bmatrix}$, $\begin{bmatrix}23&4\\28&11\end{bmatrix}$, $\begin{bmatrix}31&14\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.192.5.g.1 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: | $2^{30}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 64.2.a.a$^{3}$, 1600.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 5 y^{2} + 2 w t $ |
| $=$ | $5 z^{2} + w^{2} - t^{2}$ | |
| $=$ | $10 x^{2} + w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 100 x^{4} z^{4} - 500 x^{2} y^{4} z^{2} + 240 x^{2} z^{6} + 5625 y^{8} - 600 y^{4} z^{4} + 16 z^{8} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=17,29$, 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 8.96.3.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}-Y^{4}-Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.192.5.g.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 100X^{4}Z^{4}-500X^{2}Y^{4}Z^{2}+240X^{2}Z^{6}+5625Y^{8}-600Y^{4}Z^{4}+16Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.3-8.e.1.4 | $8$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.192.1-40.e.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.e.1.12 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.e.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.e.2.12 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.l.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.l.1.12 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.3-8.e.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.192.3-40.p.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.p.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.1.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.1.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.2.12 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.u.2.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.69-40.bf.1.8 | $40$ | $5$ | $5$ | $69$ | $13$ | $1^{26}\cdot2^{15}\cdot4^{2}$ |
| 40.2304.73-40.lq.1.14 | $40$ | $6$ | $6$ | $73$ | $6$ | $1^{28}\cdot2^{4}\cdot4^{8}$ |
| 40.3840.137-40.im.1.11 | $40$ | $10$ | $10$ | $137$ | $26$ | $1^{54}\cdot2^{19}\cdot4^{10}$ |